What Do You Expect ACE Help

What Do You Expect?

Investigation 1

1a. How many ways are there to get an even number? How many ways are there to get an odd

number? Is this equally likely?

1b. Are there an equal amount of left handed and right handed people?

1c. Try tossing a marshmallow. Does it land on its curved side more than the ends or is it equally likely?

1d. How many ways are there to get a heart? How many ways are there to get a club? How many ways

are there to get a diamond? How many ways are there to get a spade? Is this equally likely?

1e. What are possible outcomes for tossing three coins? How many ways are there to get 3 heads?

How many ways are there to get 2 heads and a tail? How many ways are there to get a head and 2 tails?

How many ways are there to get 3 tails? Are these equally likely?

2a. Theoretical probability can be written as a fraction, as follows:

For example: Find the probability of rolling a 2 with a six-sided number cube.

There are 6 outcomes that could happen. You could roll a 1, 2, 3, 4, 5 or 6. Only one of those outcomes

is a 2. The probability would be

because there is only way to get a 2 out of 6 total things that could

happen. Probability problems are written with a capitol P and then the favorable event is in

parentheses. In the example, the probability of getting a 2 would be written P(2).

Other examples are as follows:

P(3) =

P(even) =

=

P(3 or 4) =

=

Another example: There are 5 marbles in a bag. One is yellow, two are red and two are blue.

P(yellow) =

P(red) =

P(red or blue) =

How many cans are corn? How many total cans are there? What is the probability of getting corn?

2b. Theoretical probability can be written as a fraction, as follows:








P(3) =

P(even) =

=

P(3 or 4) =

=


P(yellow) =

P(red) =

P(red or blue) =

How many cans are beans? How many total cans are there? What is the probability of getting beans?

2c. Theoretical probability can be written as a fraction, as follows:








P(3) =

P(even) =

=

P(3 or 4) =

=


P(yellow) =

P(red) =

P(red or blue) =

How many cans are not spinach? How many total cans are there? What is the probability of not getting

spinach?

2d. Theoretical probability can be written as a fraction, as follows:








P(3) =

P(even) =

=

P(3 or 4) =

=


P(yellow) =

P(red) =

P(red or blue) =

How many cans are beans or tomatoes? How many total cans are there? What is the probability of

getting beans or tomatoes?

2e. Are all of the probabilities the same for each vegetable?

3a. Tree diagrams are tools to help you find outcomes. For example, if you flipped a coin 3 times and

you wanted to find all of the different outcomes, you could make a tree diagram to help you.

Make a tree diagram to find all of the possible menus.

3b. How many outcomes included a hot dog, cole slaw and an orange? How many total outcomes are

there? What is the probability of getting a hot dog, cole slaw and an orange?

3c. How many outcomes did not include a hot dog? How many total outcomes are there? What is the

probability of getting a lunch without a hot dog?

4a. What are the different outcomes? How many outcomes included loafers, blue socks and a plaid

cap? How many total outcomes are there? What is the probability of wearing loafers, blue socks and a

plaid cap?

4b. What are the different outcomes? How many outcomes included sneakers, either red or blue socks

and a green cap? How many total outcomes are there? What is the probability of wearing sneakers,

either red or blue socks and a green cap?

4c. What are the different outcomes? How many outcomes included neither red socks nor a red cap?

How many total outcomes are there? What is the probability of wearing neither red socks nor a red

cap?

5. Since the outcomes are equally likely, the spaces for blue, red, red, green, and yellow must all be the

same size. Which spinner has all of these spaces and in equal sizes?

6. Use the tree diagram to help you list the outcomes.

3rd Coin2nd Coin1st Coin

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

OU

TC

OM

ES

H

H

H

T

T

H

T

T

H

H

T

T

H

T

7. Which color and number combination occurs most often?

8. How many outcomes include red and 3? How many total outcomes are there? What is the

probability of getting red and 3?

9. How many outcomes include not a 3? How many total outcomes are there? What is the

probability of getting not a 3?

10a. How many outcomes include blue? How many total outcomes are there? What is the probability

of getting blue? How many outcomes include yellow? How many total outcomes are there? What is

the probability of getting yellow? How many outcomes include red? How many total outcomes are

there? What is the probability of getting red?

10b. You found the probabilities in problem 10a. If there are 24 contestants, how many would you

expect to get blue? Red? Yellow? Since each blue block pays $5, each red block pays $10, and each

yellow block pays $50, how much money should the game show expect to pay out?

11a. Use the table below to help you find all of the outcomes.

1 2 3 4 5 6

1 1 2 2 4

3 9 12 4

5

6

How many outcomes are prime? How many outcomes are not prime? How many total outcomes are

there? What is the probability of getting a prime number? What is the probability of not getting a

prime number? If the cubes are rolled 100 times, how many times would Player A win? How many

times would Player B win? Since Player A gets 10 points for each win, how many points would Player A

get? Since Player B gets one point for each win, how many points would Player B get?

11b. DId Player A and Player B end up with about equal points?

12. Does probability mean that is what must happen? When we say the probability of getting heads

when flipping a coin is

, does that mean that if you flip a coin twice you will only get one head? Who is

correct Rachel or Mariana? Why?

13. Use the table you created in problem 11a to help you answer this question. Explain your reasoning.

14. Use the number line below to mark each event.

Think about each event. Will it never happen (impossible)? Will it always happen (certain)? Will it

happen half of the time (0.5)?

15. Try printing the diagram below and dividing it into equal parts.

16. Could you divide the picture into more equal parts to help you find other equivalent fractions? You

could also look at these equivalent fractions to help you find a pattern in equivalent fractions.

=

=

=

=

=

=

=

=

17. Which fraction is equivalent to the shaded part?

18. Could you divide the picture into more equal parts to help you find other equivalent fractions? You

could also look at these equivalent fractions to help you find a pattern in equivalent fractions.

=

=

=

=

=

=

=

=

10 0.5

19a. Count the tally marks to find out how many times Fala spun the spinner.

19b. How many spins were blue? How many spins were yellow? Write these as fractions. Change each

fraction to a decimal by dividing the numerator by the denominator. Then change each decimal to a

percent by moving the decimal two places to the right.

19c. Since you need to answer this based on theoretical probabilities, you need to find how many equal

parts of the spinner are blue and how many equal parts are yellow. Using this information, set up the

probability of spinning blue and the probability of spinning yellow. Change these fractions to percents.

19d. Compare the percents you found in problems 19b and 19c. Are they the same? If they are

different, tell why you think they are different.

20a. How many times did the tack land point up when it was dropped 100 times? Using this

information, how many times would you expect the tack to land point up if it were dropped 500 times?

20b. Do the outcomes have the same chance to land point up and point down?

20c. Theoretical probabilities are based on equally likely outcomes. Were these outcomes equally

likely?

21a. Based on Juanitas tally marks, how many times did people win? How many total times did they

play? What is the experimental probability?

21b. Use the experimental probability you found in problem 21a to help you find how many times she

can expect to win in 20 throws.

21c. Remember, it costs one ticket to play and she gets ten tickets if she wins.

21d. Theoretical probabilities are based on equally likely outcomes. Were these outcomes equally

likely?

22. Try using the percent bars below to help you find the number of red and blue marbles.

Red Marbles

0 100

0% 50% 100%

0 100

100%0% 50%

Blue Marbles

Once you know the number of red and blue marbles, can you find the number of white marbles?

23. Hannah got how many blue out of how many draws from the bucket? What percent is this? Can

you use this percent to find out how many blocks would be blue out of 72 draws?

24. It may be helpful to use the chart below to find the total possible outcomes. Remember, you are

looking for the product of the number cubes so you need to multiply to fill in the chart.

How many outcomes are a multiple of 5? How many total outcomes are there? What is the probability?

25. Use the probability you found in problem 24 to help you find this answer. A proportion may be

helpful.

26. Use the chart you made in problem 24. How many outcomes are a multiple of 7? How many total

outcomes are there? What is the probability?

27. Use the probability you found in problem 26 to help you find this answer. A proportion might be

helpful.

28a. Use the chart you made in problem 24. How many outcomes are multiples of both 3 and 4? How

many total outcomes are there? What is the probability?

28b. Use the chart you made in problem 24. How many outcomes are multiples of either 3 and 4? How


28c. Use the chart you made in problem 24. How many outcomes have both 5 and 3 as factors? How


1 2

12

9

1 2 3 4 5 6

1

2

3

4

5

6

28d. Use the chart you made in problem 24. How many outcomes are prime numbers? How many

total outcomes are there? What is the probability?

28e. Use the chart you made in problem 24. How many outcomes are greater than 10? How many

total outcomes are there? What is the probability?

28f. Use the chart you made in problem 24. How many outcomes are less than 18? How many total

outcomes are there? What is the probability?

29. In a tree diagram, every set of branches must be equally likely within each set. Is that true in this

tree diagram?

30. What are the outcomes? What is the probability of getting an even sum? What is the probability of

getting an odd sum? Are these sums equally likely? Is it a fair game? Dont forget to explain your

answer.

31. Since probability is determined with outcomes that are equally likely, divide each board into equal

parts. For each board, how many parts are marked A (blue)? How many total parts are there? What is

the probability of hitting A (blue)? For each board, how many parts are marked B (red)? How many

total parts are there? What is the probability of hitting B (red)? For each board, how many parts are

marked C (yellow)? How many total parts are there? What is the probability of hitting C (yellow)?

32. Since probability is determined with outcomes that are equally likely, divide each board into equal

parts (you may have already done this in problem 31). How many parts are marked A (blue) or B (red)?

How many total parts? What is the probability of hitting A or B?

33. What parts are not C? Since probability is determined with outcomes that are equally likely, divide

each board into equal parts (you may have already done this in problem 31). How many parts are

marked A (blue) or B (red)? How many total parts? What is the probability of hitting A or B?

Investigation 2

1. How many outcomes are there? How many ways are there to make purple? What is the probability

of making purple? How many ways are there to NOT get purple? What is the probability of not getting

purple? Using the probabilities, how many chances does Bonita have to win? Use this amount to help

you decide how many points Deoin and Bonita should score with each win.

2a. Where is Bag 1 in the chart? What colors are in Bag 1?

2b. Where is Bag 2 in the chart? What colors are in Bag 2?

2c. Use the area model below to help you.

2d. How many total outcomes are there in the chart? How many outcomes include red-blue or blue-

red? What is the probability of winning?

3a. Use the area model below to help you.

Red -

White

Bag 2

Ba

g 1

Red Blue

Wh

ite

Ho

t-p

ink

Ne

on

-

Ye

llo

w

Bin

1

Bin 2

How many outcomes are there? How many ways can Kira pick a neon-yellow toothbrush and grape

gum? What is the probability of choosing a neon-yellow toothbrush and grape gum?

3b. Use the probability you found in problem 3a to help you find this answer.

4a. What trails can Al choose at the top of the mountain? How can you use a number cube to help him

make the decision of which trail to take? Half way down the mountain, Al needs to make another choice

of trails. How can you use the number cube to help Al choose which trail to take in each case?

4b. Play the game you designed and collect the data to help you determine the experimental

probability. How many total outcomes are there? How many times did you end up at the lodge? How

many times did you end up at the lift? How many times did you end up at the ski shop? What is the

experimental probability for each of these destinations?

4c. How many total routes are there? How many routes lead to the lodge? What is the probability of

ending up at the lodge?

5a. How many different paths are there? How many paths led to cave A? What is the theoretical

probability of ending at Cave A? How many paths lead to Cave B? What is the theoretical probability of

ending at Cave B?

5b. Use the probabilities you found in problem 5a to help you find how many times out of 100 you

would expect to end in Cave A and Cave B.

6. Split the sections into equal sized spaces. How much space out of the total space ends in Cave A?

How much space out of the total space ends in Cave B?

7. Find where you can end up with any choice of paths. Which map shows a route where you end up in

A, or if you go the other way, you can end up in A or B with either original choice?

8a. How many marbles and what colors are possible for the first draw? How would you show all of

these choices are possible for the first draw in a tree diagram? How many marbles and what colors are

possible for the second draw? How would you show all of these choices are possible for the second

draw in a tree diagram?

8b. What is one possible marble you could choose in the first draw? What are all of the possible

combinations with that marble as the first draw? What is a different possible marble for the first draw?

What are all of the possible combinations with that marble? Repeat this process until you have used all

of the possible first draw marbles.

8c. An area model is started for you below.

orange orange blue

orange

blue

8d. A chart would be just like an area model only using letters instead of words.

9. How many outcomes are the same color? How many total outcomes? What is the probability of

getting the same color? Use this probability to help you predict how many times out of 50 you will

probably get the same color.

10. What is the probability of getting a match? What is the probability of not getting a match? How

many chances out of 25 will result in a match? How many chances out of 25 will result in not getting a

match? Use this information to help you assign points for each outcome.

11. Find all of the ways the marbles can be arranged. Then find the probability of choosing a green

marble. Which arrangement has the greatest probability of choosing green?



13a. How many seniors favor the rule? How many total seniors were surveyed? What is the probability

of a senior favoring the rule?

13b. How many seniors drive to school and favor the rule? How many total seniors were surveyed?

What is the probability that a senior drives to school and favors the rule?

13c. How many seniors drive to school and oppose the rule? How many total seniors were surveyed?

What is the probability that a senior drives to school and opposes the rule?

13d. Are seniors the only ones affected by this rule? How do you think other students might react? Do

you think others would be in favor or oppose the rule?

14a. Try using a tree diagram, area model or organized list to help you find all of the outcomes.

14b. How many ways are there to get a negative score? How many total outcomes? What is the

probability of getting a negative score? How many ways are there to get a positive score? How many

total outcomes? What is the probability of getting a positive score? Are the probabilities equal?

15. How many squares are in the library? How many total squares are there? What is the probability

that the treasure is in the library?

16. How many squares are in the den? How many total squares are there? What is the probability that

the treasure is in the den?

17. How many squares are in the dining hall? How many total squares are there? What is the

probability that the treasure is in the dining hall?

18. How many squares are in the great hall? How many total squares are there? What is the

probability that the treasure is in the great hall?

19. How many squares are in the front hall? How many total squares are there? What is the probability

that the treasure is in the front hall?

20. What happens to the total area if the floor plan is enlarged by a scale factor of 2? What happens to

the total area if each room is enlarged by a scale factor of 2? What happens to the probabilities? If you

arent sure, try enlarging the floor plan on graph paper and then finding the probabilities.

21. How many total squares would need to be on the floor plan? How many squares would need to be

in each room? Design a floor plan that meets these requirements.

22a. Divide the dartboard into equal parts. How many parts are marked A? How many total parts on

the board? What is the probability of landing on A?

22b. Divide the dartboard into equal parts. How many parts are marked B? How many total parts on

the board? What is the probability of landing on B?

23a. Divide Dartboard 1 into equal parts. How many parts are marked A? How many total parts on the

board? What is the probability of landing on A? Divide Dartboard 2 into equal parts. How many parts

are marked A? How many total parts on the board? What is the probability of landing on A?

23bi. It costs $1 to play and you win $2 each time you land on B. How much will it cost you to play 36

times? What is the probability of landing on B on Dartboard 1 in 36 chances? How much can you expect

to win out of 36 chances? Subtract the cost to play from your winnings. Did you win or lose overall?

How much did you win or lose?

It costs $1 to play and you win $2 each time you land on B. How much will it cost you to play 36

times? What is the probability of landing on B on Dartboard 2 in 36 chances? How much can you expect

to win out of 36 chances? Subtract the cost to play from your winnings. Did you win or lose overall?

How much did you win or lose?

23bii. If you win, the carnival loses that much money. If you lose, the carnival wins that much money.

Dont forget to answer for both dartboards.

23c. Which game(s) has a probability in favor of the carnival?

24a. Use the chart below to help you find all of the outcomes.

1 2 3 4 5 6

1 1, 1 1, 2 1, 3

2

3 3, 2

4 4, 5

5

6

What are the factors of 5? Circle the outcomes that include only the factors of 5. How many outcomes

included only factors of 5? How many total outcomes? What is the probability of getting only factors of

5?

24b. Will anything change if you roll two cubes once rather than one cube twice? What is the

probability of getting only factors of 5 with two cubes?

24c. Were the probabilities you found in problems 24a and 24b the same or different? Why did this

happen?

25. What are the outcomes if you spin Spinner A twice? How many outcomes are red and blue in any

order? How many total outcomes? What is the probability of getting red and blue (in any order)? What

are the outcomes if you spin Spinner B twice? How many outcomes are red and blue in any order? How

many total outcomes? What is the probability of getting red and blue (in any order)? What are the

outcomes if you spin Spinner A once and Spinner B once? How many outcomes are red and blue in any

order? How many total outcomes? What is the probability of getting red and blue (in any order)?

Which way gave you the greatest chance of winning?



27. Find all of the ways the marbles can be arranged. Then find the probability of choosing a red

marble. Which arrangement has the greatest probability of choosing red?

28. Look at problems 11, 12, 26, and 27 to give you ideas. Find all of the ways the marbles can be

arranged. Then find the probability of choosing a winning marble. Which arrangement has the greatest

probability of choosing a winning marble?

Investigation 3

1. A one-and-one free throw situation means the player gets to make a shot. If the player makes the

shot, they get to try again for another shot. If they miss, they dont get the second shot. Each shot is

worth one point. If they miss the first shot, they get zero points. If they make the first shot but miss the

second shot, they get one point. If they make both shots, they get two points. If a player had a 70%

average, they would probably make

shots, so the grid would look as follows:

They only get to shoot the second shot if they make the first shot, so the grid for the second shot would look like this:

In this case, the player has an 80% average. What would the grid look like?

2. A one-and-one free throw situation means the player gets to make a shot. If the player makes the

shot, they get to try again for another shot. If they miss, they dont get the second shot. Each shot is

worth one point. If they miss the first shot, they get zero points. If they make the first shot but miss the

second shot, they get one point. If they make both shots, they get two points. If a player had a 70%

average, they would probably make


30

%7

0%1

st

sh

ot

1s

t s

ho

t

30

%7

0%

70% 30%

2nd shot

0 points

1 p

oin

t

2 points


In this case, the player has an 40% average. What would the grid look like?

3a. Try using fractions, decimals or percentages to compare the statistics.

3b. Make a fraction to represent each players probability. Change each fraction to a percent.

4a. What was Alexs free throw percentage (you found this in problem 3b)? A one-and-one free throw

situation means the player gets to make a shot. If the player makes the shot, they get to try again for

another shot. If they miss, they dont get the second shot. Each shot is worth one point. If they miss

the first shot, they get zero points. If they make the first shot but miss the second shot, they get one

30

%7

0%1

st

sh

ot

1s

t s

ho

t

30

%7

0%

70% 30%

2nd shot

0 points

1 p

oin

t

2 points

point. If they make both shots, they get two points. If a player had a 70% average, they would probably

make



In this case, use Alexs free throw percentage. What would the grid look like?

4b. How many squares in the hundred grid represented zero points? How many squares in the hundred

grid represented one point? How many squares in the hundred grid represented two points?

4c. Take the total number of points and divide by 100.

4d. What was Gerrits free throw percentage (you found this in problem 3b)? A one-and-one free throw

situation means the player gets to make a shot. If the player makes the shot, they get to try again for

30

%7

0%1

st

sh

ot

1s

t s

ho

t

30

%7

0%

70% 30%

2nd shot

0 points

1 p

oin

t

2 points

another shot. If they miss, they dont get the second shot. Each shot is worth one point. If they miss

the first shot, they get zero points. If they make the first shot but miss the second shot, they get one

point. If they make both shots, they get two points. If a player had a 70% average, they would probably

make



In this case, use Gerrits free throw percentage. What would the grid look like?

How many squares in the hundred grid represented zero points? How many squares in the hundred grid

represented one point? How many squares in the hundred grid represented two points?

Take the total number of points and divide by 100.

30

%7

0%1

st

sh

ot

1s

t s

ho

t

30

%7

0%

70% 30%

2nd shot

0 points

1 p

oin

t

2 points

5a. What is Gerrits free throw percentage? How would you show the first shot in a hundred grid? How

would you show the second shot in the hundred grid? Which area shows two points? What percent is

this? Which areas show one point? What percent is this? Which area shows zero points? What

percent is this?

5b. What was different about this situation?

6a. How would you show the first shot in a hundred grid? How would you show the second shot in the

hundred grid? Which area shows two points? Which area shows one point? Which area shows zero

points? Which area is largest?

6b. How many squares represented two points on the hundred grid? How many total points is this?

How many squares represented one point on the hundred grid? How many total points is this? How

many total points are represented on the hundred grid? Divide this amount by one hundred to find the

average points.

7a. How would you show the first shot in a hundred grid? How would you show the second shot in the



How many squares represented two points on the hundred grid? How many total points is this? How

many squares represented one point on the hundred grid? How many total points is this? How many

total points are represented on the hundred grid? Divide this amount by one hundred to find the

average points.

7b. How would you show the first shot in a hundred grid? How would you show the second shot in the



How many squares represented two points on the hundred grid? How many total points is this? How

many squares represented one point on the hundred grid? How many total points is this? How many

total points are represented on the hundred grid? Divide this amount by one hundred to find the

average points.

8a. Find all of the outcomes. There should be six outcomes. Find the probability of getting $11. Find

the probability of getting $2. Find the total amount of money possible using these probabilities. Then

divide by the denominator in the problem to get the expected value. Is this amount better or worse

than $10 per week? If it is better, accept the customers offer. If it is worse, reject the offer.

8b. Find all of the outcomes. Find the probability of getting $30. Find the probability of getting $2.

Find the total amount of money possible using these probabilities. Then divide by the denominator in

the problem to get the expected value. Is this amount better or worse than $10 per week? If it is

better, accept the offer. If it is worse, reject the offer.

8c. Find all of the outcomes. Find the probability of getting $50. Find the probability of getting $2. Find

the total amount of money possible using these probabilities. Then divide by the denominator in the

problem to get the expected value. Is this amount better or worse than $10 per week? If it is better,

accept the offer. If it is worse, reject the offer.

9. How many sections are labeled Bankrupt? How many total sections? What is the probability that a

player will land on bankrupt?

10. How many sections are labeled with $500 or larger than $500? How many total sections? What is

the probability that a player will get $500 or more?

11. How many sections are labeled $350? How many total sections? What is the probability that a

player will land on $350?

12. What is 10% of 90? (Hint: Move the decimal one place.) 30% is three times bigger than 10%, so

multiply the amount that is 10% of 90 by 3.

13. 25% is one-fourth. What is

of 80?

14. What is 10% of 180? 5% is half of 10%, so what is 5% of 180? 40% is four times bigger than 10%, so

what is 40% of 180? If you know 40% and 5%, could you find 45% of 180? Another way to do this is to

find 50% (50% is half) and then take 5% (remember that 5% is half of 10%) away from this amount.

15a. Try using an area model like was used in class to find Nishis free throw probabilities in

Investigation 3.1. Use Saturday for one side and Sunday for the other side of the area model. If there is

a 30% chance of rain on Saturday, how would you show this in the grid? If there was a 30% chance of

rain on Sunday, how would you show this in the hundred grid? You should end up with four sections.

Which section represents rain on both days? What percent is this?

15b. Was the percent you found in problem 15a large or small? If it is a large percent, Wanda should

have predicted the weather better. If it is a small percent, Wanda probably couldnt have been more

accurate.

15c. Try using an area model like was used in class to find Nishis free throw probabilities in

Investigation 3.1. Use Monday for one side and Tuesday for the other side of the area model. If there is

a 20% chance of rain on Monday, how would you show this in the grid? If there was a 20% chance of

rain on Tuesday, how would you show this in the hundred grid? You should end up with four sections.

Which section(s) represents rain on one or both days? What percent is this?

16. We can assume every group of 500 fish will have 150 salmon. How many groups of 500 are in

10,000? If each group has 150 salmon, how many salmon would this be?

17a. You have already found these answers in class for Investigation 3.2, problem B.

17b. How many average points occur with a 40% probability (see the chart you made in problem 17a)?

How many average points occur with an 80% probability (see the chart you made in problem 17a)? Is

this decimal twice as much as the average for 40%?

17c. How many average points occur with a 50% probability (see the chart you made in problem 17a)?

How many average points occur with an 100% probability (see the chart you made in problem 17a)? Is

this decimal twice as much as the average for 50%?

17d. Is there a steady change in the table for problem 17a? Is there a steady change in the table for

problem 17d?

18a. What are all of the possible outcomes? Which of these outcomes include at least one sum that is a

positive number? What is the probability of getting a positive number?

18b. Find the average of all of the sums from problem 18a. Averages are a way of evening out the

data. One way to find an average is to share the amounts equally. For example, if you have 3, 5, -2, 1,

-6, 4, -3, and 6 you could draw these with blocks on a number line like below:

Then even out the blocks so each pile has about the same amount. Move 3 blocks to -3.

0

Move 5 blocks to -6.

Move 2 of the blocks from 4 to -2.

0 X

0 X X

0 X X X

Move one block from 1 to -1.

We have 8 blocks left to share with 8 spaces, so each space gets one.

This is as even as we can get, so the average is 1.

Another way to do this, is to put all of the blocks together in a pile (add them) and then pass them out

evenly (divide by how many numbers you had).

3 + 5 + -2 + 1 + -6 + 4 + -3 + 6 = 8

8 8 = 1

The average is one.

19a. Find all of the possible outcomes. Multiply the first spin by the second spin in each outcome. How

many products were positive? How many total outcomes? What is the probability of getting a positive

number?

19b. Find the average of all of the products from problem 18a. Averages are a way of evening out the

data. One way to find an average is to share the amounts equally. For example, if you have 3, 5, -2, 1,

-6, 4, -3, and 6 you could draw these with blocks on a number line like below:

0 X X X X X

0

Then even out the blocks so each pile has about the same amount. Move 3 blocks to -3.

Move 5 blocks to -6.

0

0 X

Move 2 of the blocks from 4 to -2.

Move one block from 1 to -1.

We have 8 blocks left to share with 8 spaces, so each space gets one.

This is as even as we can get, so the average is 1.

0 X X

0 X X X

0 X X X X X

0

Another way to do this, is to put all of the blocks together in a pile (add them) and then pass them out

evenly (divide by how many numbers you had).

3 + 5 + -2 + 1 + -6 + 4 + -3 + 6 = 8

8 8 = 1

The average is one.

20a. The probability that Fred wins is

. This means he will probably win

of the rounds. If they play

12 rounds how many rounds will Fred probably win? Fred gets 3 points when he wins a round. How

many points should Fred expect out of 12 rounds? The probability that Joseph wins is

. This means he

will probably win

of the rounds. If they play 12 rounds how many rounds will Joseph probably win?

Joseph gets 2 points when he wins a round. How many points should Joseph expect out of 12 rounds?

20b. How many points can Fred expect out of 12 rounds (see problem 20a)? Divide this by the 12

rounds to find the points per game. How many points can Joseph expect out of 12 rounds (see problem

20a)? Divide this by the 12 rounds to find the points per game.

20c. Are the points per round (see problem 20b) the same for each player? Is this fair? Would they

both have an equally likely chance to win?

21a. Try setting up an area model using a hundred grid. Put one test on the left side and the other test

on the top. Divide each side to show the 20% and 80% results. Shat test result does each of the four

sections show? What percent shows both tests negative?

21b. Where in the area model does the test show at least one positive test? What is the total percent

for all of these areas?

22a. If the probability that the dot will land in region A is 40%, what percednt is left for the other

regions?

22b. Try using a hundred grid. Mark off 40% for region A and label this region A. You found the

percentage for the remaining regions in problem 22a. You also know regions B, C, and D have the same

probability. Therefore, they must be equal in size. Split the percentage for regions B, C, and D evenly to

find the amount of space for each of these regions. Mark these regions on your dartboard and label

them as region B, region C, and region D.

22c. Try using the degrees of a circle to help you divide the circular dartboard. How many degrees are

in any circle? What is 10% of these degrees? What is 40% of these degrees? Divide the remaining

degrees equally.

23. Here is an example using Gerrit. He made

free throws. This is

=

=

= 50%. His first shot is

50% or 0.5. His second shot is 50% of 0.5 or 0.25. His third shot is 50% of 0.25 or 0.125. Therefore, his

probability is 12.5%.

24. Try using an area model to show both free throws. David makes

free throws. What percentage is

this (round to the nearest percent)? Show this percent on the left side of the area model for one of the

free throws and on top for the other free throw. Which section shows where he makes both free

throws? What percentage is this?

25. What is the probability that Emilio will make three in a row? To help you find this, see problem 23.

How many different outcomes are there for 5 free throws? Of those outcomes, how many outcomes

will allow him to play all season? What is this probability? Which probability is greater?

26a. Here is an example using Gerrit. He made


=

=




26b. How many different outcomes are there for 4 free throws? Of those outcomes, how many allow

Curt to make at least 3 out of 4 free throws? What is this probability?

27a. Assume the player chose orange. How many spaces are orange? How many total spaces are

there? What is the probability?

27b. You found the probability of winning in problem 27a. If the player gets $2 for each of those wins in

38 spins, how much money can the player expect in 38 spins? Divide this amount by 38 to get the

expected value per spin. Since the game costs $1, does the player win or lose per spin?

28a. If Luis has an 80% free throw average, would the piece which represents making all 3 baskets be a

large piece or a small piece? Making 2 baskets would be represented by the next sized pieces. Making

one basket would be represented by the next sized pieces. Making none of the baskets would be

represented by the final piece.

28b. Here is an example using Gerrit. He made


=

=




Investigation 4

1a. Find all of the outcomes for tossing a coin 3 times. How many of these outcomes end up with heads

two or more times? What is this probability? If he plays 80 times, how many can he expect to win? If

he gets 10 tickets for each win, how many tickets can he expect in 80 plays? It costs 6 tickets to play. If

he plays 80 times, how much will it cost him to play 80 times? Does he win or lose tickets?

1b. How many tickets did he win or lose in 80 plays (You found this in problem 1a)? Divide this by the

number of plays (80). How many tickets can he expect to win or lose for each play?

2a. Try finding the outcomes for each situation. Are the outcomes the same or different?

2b. What is the probability of getting 3 tails? Does the probability change in any way from the first toss

to the second toss?

3a. Try making a tree diagram or an organized list to help you find all of the outcomes.

3b. Using the outcomes you found in problem 3a, what is the probability of 4 male puppies? Which

probability is greater?

4. Use the outcomes you found in problem 3a. How many outcomes showed 4 females? How many

total outcomes? What is the probability of getting 4 females?

5. Use the outcomes you found in problem 3a. How many outcomes showed 2 males and 2 females?

How many total outcomes? What is the probability of getting 2 males and 2 females?

6. Use the outcomes you found in problem 3a. How many outcomes showed at least one male? How

many total outcomes? What is the probability of getting at least one male?

7. Use the outcomes you found in problem 3a. How many outcomes showed at least one female? How

many total outcomes? What is the probability of getting at least one female?

8. Is it equally likely to have a male or female? What is the probability of getting a male? What is the

probability of getting a female? What is the average amount per puppy? If she has 4 puppies, how

much can she expect to make?

9. You found the average amount per puppy in problem 8. What do you need to do with this average

amount to find how much she can expect to make from selling 5 puppies?

10a. List all of the possibilities. Then decide who won for each possibility and how many games had to

be played. In how many ways was the series finished in 3 games? How many total possibilities? What

was the probability that the series ended in 3 games? In how many ways was the series finished in 4

games? How many total possibilities? What was the probability that the series ended in 4 games? In

how many ways was the series finished in 5 games? How many total possibilities? What was the

probability that the series ended in 5 games?

10b. How many ways showed a win for the Stars? How many total possibilities? What is the probability

that the Stars win?

11a. Use a tree diagram. What the possible outcomes for the first toss? What are the possible

outcomes for the second toss? How many outcomes so far? What are the outcomes for the third toss?

How does the fact that there are two outcomes (H and T) for flipping one coin and 3 different tosses

help you find the total outcomes?

11b. Use what you learned about the number of outcomes for one toss and the number of tosses to

help you find the total outcomes for 4 and 5 tosses. If this doesnt help, extend the tree diagram to find

the number of outcomes.

11c. If you arent sure, list the possibilities (or use your list from problem 11b if you did this in that

problem).

11d. Symmetry is when you have a mirrored image. If you dont see the symmetry, try putting your

possibilities in alphabetical order.

12. Use what you learned about 5 heads tossed in problem 11c.

13a. Try making a tree diagram that shows curl or no-curl (just like you did for heads and tails in

problem 11). How many ways show two children that can curl their tongues? How many total

possibilities? What is the probability that both children can curl their tongues?

13b. Try making a tree diagram for curl or cant curl. How many possibilities show the child being

able to curl their tongue? How many total possibilities are there? What is the probability that the

children will be able to curl their tongues?

13c. Use the tree diagram you made in problem 13b. How many possibilities show curl for the first

child? How many total possibilities are there? What is the probability that only the oldest child will be

able to curl their tongue?

14a. Try making a chart like below:

All Closed One Open Two Open Two Closed One Closed All Open

CCCCC CCCCO CCCOO CCOOO C0000 OOOOO

CCCOC CCOCO COCOO OCOOO

How many different routes are there? Then decide for each route if you would be able to reach the

castle. (CCOOO means only gates 3, 4, and 5 are open). Can you get from one castle to the other with

only those gates open? (Yes, gate 5, then 4 gets you to the castle.) How many routes work? How many

total routes? What is the probability that a route is open?

14b. How many outcomes were in each situation? What kind of probability is this?

15a. What are the outcomes for tossing the coins? How much money would be collected for each of

these outcomes? What is the average amount of money collected for these outcomes? How does this

compare with collecting $10 per week? Is this a fair deal?

15b. What are the outcomes for tossing the coins? How much money would be collected for each of

these outcomes? What is the average amount of money collected for these outcomes? How does this

compare with collecting $10 per week? Is this a fair deal?

16a. What are the possible outcomes of flipping a coin three times? Where would the marker be at the

end of the turn for each of the outcomes?

16b. What are the possible outcomes of flipping a coin four times? Where would the marker be at the

end of the turn for each of the outcomes?

17a. What colors can the pointer land on? How much of the circle is covered for each of these

outcomes?

17b. Does spinning the spinner once change what could happen if you spin it again? Find all of the

outcomes of spinning the spinner three times. How many ways end up with RBB? How many total

outcomes? What is the probability of getting RBB?

17c. What are the possible colors the pointer could land on? How many ways end up with each

outcome? How many total outcomes? What is the probability of getting each outcome? (There are

several answers needed here.)

17d. A binomial situation means there are two possible outcomes. Are there two possible outcomes

with this spinner?

17e. What are the possible outcomes when you spin the spinner three times? How many total

outcomes? How many ways end up with RBB? What is the probability of getting RBB?

18. Do you notice anything interesting in the triangle? Does anything repeat in a similar manner?

Describe the patterns you see.

19. Each number in the triangle is found by adding the numbers diagonally above it. Create the next

row (6th row) in the triangle. The numbers in each row refer to the different outcomes that are possible.

20. Could you use Pascals triangle to find the probability? If not, find all of the different outcomes.

Then find how many outcomes have at least two heads. What is the probability?

21. Could you use Pascals triangle to find the probability? If not, find all of the different outcomes.

Then find how many outcomes have at least two heads. What is the probability?

22. Try extending Pascals triangle to nine rows. How many total outcomes are shown in this row?

Which number would represent exactly three correct answers? What would the probability be?

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