COUNTING OUTCOMESPERMUTATIONS & COMBINATIONS

PROBABILITYREVIEW

COUNTING OUTCOMES

Andy has 3 pairs of pants: 1 gray, 1 blue and 1 black. He has 2 shirts: 1 white and 1 red. If Andy picks 1 pair of pants and 1 shirt, how many different outfits does he have?

Andy can choose 1 of 3 pairs of pants and 1 of 2 shirts. A tree diagram can help you count his choices.

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The total number of choices is the product of the number of choices for item A, the number of choices for item B, etc. You can also use the counting principle.

FIRST ● SECOND ● THIRD, etc. = TOTAL NUMBER OF CHOICES

FIND THE TO TAL N UMBER O F CHO ICES.

1. Moesha has 6 pairs of socks and 2 pairs of sneakers. She chooses 1 pair of socks and 1 pair of sneakers. How many possible combinations are there?

2. Kim has 5 swimsuits, 3 pairs of sandals, and 2 beach towels. In how many ways can she pick one of each to go to the beach with?

PERMUTATIONS

The expression 5! is read “5 factorial”.

It means the product of all whole numbers from

5 to 1. 5! =

Evaluate

!3!5

How many 3-letter codes can be made from A, B, C, D, E, F, G, H with no repeating letters?

This is a permutation problem. ORDER IS IMPORTANT. ABC is different from ACB.

• There are choices for the first letter.• There are choices for the

second letter.• There are choices for the third

letter.

PERMUTATION FORMULA

n is the number of objects and r is the number chosen.

rnP

You can write the code as , meaning the number of permutations of 8 objects chosen at 3 times.

38P

The number of codes possible x x =

Evaluate each factorial.

Find the value of each expression.

1. !4 2. !7 3.!3!8 4. !2 !5

5. 36 P 6. 25 P 7. 312P 8. 415P

Solve.

9. In how many ways can you pick a football center and quarterback from 6 players who try out? 10. For a meeting

agenda, in how many ways can you schedule 3 speakers out of 10 people who would like to speak?

COMBINATIONSMr. Jones wants to pick 2 students from Martin, Joan, Bart, Esperanza, and Tina to demonstrate an experiment. How many different pairs of students can he choose?

In this combination problem, the ORDER DOES NOT MATTER. What are the possibilities?

There are possible combinations.

COMBINATION FORMULA

n is the number of objects and r is the number chosen

The number of combinations of 5 students taken 2 at

a time is:

25C

!rPC rn

rn

FIN D THE NU MBER O F COM BIN ATIO NS:

1. 36C 2. 49C

3. 57C 4. 34C

Solve.

5. In how many ways can Susie choose 3 of 10 books to take with her on a trip?

6. In how many ways can Rosa select 2 movies to rent out of 6 that she likes?

Probability: • Notation: P(event)

Theoretical Probability:• The likelihood of an event occurring.• Equation: # of favorable outcomes

# of total outcomes

Experimental Probability: • The number of times an event occurs in an

experiment. • Equation: # of trials an outcome occurs

total # of trials

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An event who’s outcome is NOT based on a previous outcome.

DEPENDENT EVENT

An event who’s outcome is based on a previous outcome.

INDEPENDENT EVENT

Draw a card, keep it, then drawing another card.

Spinning a spinner, and then rolling a dice

WITH REPLACEMENTP(A and B) = P(A) • P(B)

A bag of marbles contains 6 blue, 5 red, 3 green, 4 orange, and 2 purple. You draw a marble at random, record your findings, replace the marble, then draw again.

FindEx) P(blue, blue) = You Trya) P(purple, orange) = b) P(black, blue) =

1009

103

103

WITHOUT REPLACEMENTP(A and B after A) = P(A) • P(B after A)

FindEx) P(yellow, pink) =

You Trya) P(black, black) = b) P(Nike, pink) =

953

196

101

A sock drawer contains 4 black, 2 yellow, 3 polk-a-dot, 5 Nike, and 6 pink. You pick a sock at random, record your findings, then pick another without replacing the first.