Unit 1 Review (4.1-4.5)MDM 4U

Mr. Lieff

Test Format 20 MC (4 per section) 15 Marks K/U 20 Marks APP (choice 3 of 5) 10 Marks TIPS (choice 2 of 3) 15% COMM (using formulas, good form,

rounding, concluding statements)

4.1 Intro to Simulations and Experimental Probability Design a simulation to model the probability of an

event Ex: design a simulation to determine the

experimental probability that more than one of 5 keyboards chosen in a class will be defective if we know that 25% are defective

1. Get a well-shuffled deck of cards, choosing clubs to represent a defective keyboard

2. Choose 5 cards with replacement and record how many are clubs

3. Repeat a large number of times (e.g., 4 outcomes x 10 = 40) and calculate the experimental probability

4.2 Theoretical Probability

Work with Venn diagrams ex: Create a Venn diagram illustrating the

sets of face cards and red cards S = 52red & face = 6

red = 20face = 6

4.2 Theoretical Probability

Calculate the probability of an event or its complement

Ex: What is the probability of randomly choosing a male from a class of 30 students if 10 are female?

P(A) = n(A) = 20 = 0.67 n(S) 30

4.2 Theoretical Probability Ex: Calculate the probability of not throwing a

total of four with 3 dice There are 63 = 216 possible outcomes with three

dice Only 3 outcomes produce a 4 P(sum = 4) = 3_

216 probability of not throwing a sum of 4 is:

1 – 3_ = 213 = 0.986 216 216

4.3 Finding Probability Using Sets

Recognize the different types of sets (union, intersection, complement)

Utilize the Additive Principle for the Union of Two Sets: n(A U B) = n(A) + n(B) – n(A ∩ B) P(A U B) = P(A) + P(B) – P(A ∩ B)

4.3 Finding Probability Using Sets

Ex: What is the probability of drawing a red card or a face card

Ans: P(A U B) = P(A) + P(B) – P(A ∩ B) P(red or face) = P(red) + P(face) – P(red and face) = 26/52 + 12/52 – 6/52

= 32/52 = 0.615

4.3 Finding Probability Using Sets

What is n(B υ C) 2+8+3+3+6+2+1+8+1 = 34 What is P(A∩B∩C)? n(A∩B∩C) = 3 = 0.07 n(S) 43

4.4 Conditional Probability 100 Students surveyed

Course Taken No. of students

English 80

Mathematics 33

French 68

English and Mathematics

30

French and Mathematics

6

English and French

50

All three courses 5

What is the probability that a student takes Mathematics given that he or she also takes English?

4.4 Conditional Probability

M

F

E

5

25

45

5

1

2

17

4.4 Conditional Probability

To answer the question in (b), we need to find P(Math|English).

We know... P(Math|English) = P(Math ∩ English)

P(English) Therefore…

P(Math|English) = 30 / 100 = 30 x 100 = 3

80 / 100 100 80 8

4.4 Conditional Probability

Calculate the probability of event B occurring, given that A has occurred Need P(B|A) and P(A)

Use the multiplicative law for conditional probability

Ex: What is the probability of drawing a jack and a queen in sequence, given no replacement?

P(1J ∩ 2Q) = P(2Q | 1J) x P(1J) = 4 x 4 = 16 = 0.006 51 52 2 652

4.5 Tree Diagrams and Outcome Tables

A sock drawer has a red, a green and a blue sock You pull out one sock, replace it and pull another out Draw a tree diagram representing the possible outcomes What is the probability of drawing 2 red socks? These are independent events

R

R

R

R

B

B

B

BG

G

G

G

91

31

31

)()()(

redPredPredandredP

4.5 Tree Diagrams and Outcome Tables Mr. Lieff is going fishing He finds that he catches fish 70% of the time

when the wind is out of the east He also finds that he catches fish 50% of the

time when the wind is out of the west If there is a 60% chance of a west wind

today, what are his chances of having fish for dinner?

We will start by creating a tree diagram

4.5 Tree Diagrams and Outcome Tables

west

east

fish dinner

fish dinner

bean dinner

bean dinner

0.6

0.4 0.7

0.3

0.5

0.5

P=0.3

P=0.3

P=0.28

P=0.12

4.5 Tree Diagrams and Outcome Tables P(east, catch) = P(east) x P(catch | east) = 0.4 x 0.7 = 0.28 P(west, catch) = P(west) x P(catch | west) = 0.6 x 0.5 = 0.30 Probability of a fish dinner: 0.28 + 0.3 = 0.58 So Mr. Lieff has a 58% chance of catching a

fish for dinner

Review

Read class notes and home learning Complete

pp. 268-269 #3-4, 5abceg, 7, 9, 10; p. 270 #1-3, 7 pp. 324-325 #1, 2; p. 326 #1