Warm up A ferris wheel holds 12 riders. If there are 20 people waiting to ride it, how many ways can they ride it?

Warm up

A ferris wheel holds 12 riders. If there are 20 people waiting to ride it, how many ways can they ride it?

Solution

Since only 12 of the 20 people can ride the ferris wheel at a time, there are C(20,12) or 125 970 different groups of riders.

Each group can be placed on the ferris wheel 11! or 39 916 800 ways since it is a circular permutation.

So the total number of ways is:

125 970 x 39 916 800 = 5 028 319 296 000

or 5.03 x 1012

I hope they purchased the season pass!

Chapter 4 ReviewMDM 4U

Mr. Lieff

4.1 Intro to Simulations and Experimental Probability be able to design a simulation to investigate

the experimental probability of some 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 shuffled deck of cards, choosing clubs to represent a defective keyboard

2. choose 5 cards with replacement and see how many are clubs

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

4.2 Theoretical Probability

work effectively with Venn diagrams ex: create a Venn diagram illustrating the sets

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

red = 20face = 6


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)÷n(S) = 20÷30 = 0.67


ex: calculate the probability of not throwing a total of four with 3 dice

there are 63 possible outcomes with three dice

only 3 outcomes produce a 4 probability of a 4 is:

3/63

probability of not throwing a sum of 4 is: 1- 3/63 = 0.986

4.3 Finding Probability Using Sets

recognize the different types of sets utilize the additive principle for unions of sets 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)

calculate probabilities using the additive principle


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


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 surveyedCourse 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?


M

F

E

5

25

45

5

1

2

17


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 / 100 100 808


calculate a probability of events A and B occurring, given that A has occurred

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(J ∩ Q) = P(Q | J) x P(J) = 4/51 x 4/52 = 16/2652 = 0.006

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

9

1

3

1

3

1

)()(

)(

redPredP

redandredP

4.5 Tree Diagrams and Outcome Tables Mr. Greer 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. Greer has a 58% chance of catching a

fish for dinner

4.6 Permutations

find the number of outcomes given a situation where order matters

calculate the probability of an outcome or outcomes in situations where order matters

recognizing how to restrict the calculations when some elements are the same

4.6 Permutations

ex: How many ways can 5 students be arranged in a line?

ans: 5! = 120 ex: How many ways are there if Jake must be first? ans: (5-1)! = 4! = 60 ex: in a class of 10 people, a teacher must pick 3 for

an experiment (students are tested in a particular order)

How many ways are there to do this? ans: P(10,3) = 10!/(10 – 3)! = 720

Permutations cont’d

How many ways are there to rearrange the letters in the word TOOLTIME?

8! / (2!2!) = 8! / (4) = 10 080

4.6 Permutations

ex: what is the probability of opening one of the school combination locks by chance? Second digit must be different from the first

ans: 1 in 60 x 59 x 59 = 1 in 208 860 Circular Permutations: (n-1)! ways to arrange

n objects in a circle.

4.7 Combinations

find the number of outcomes given a situation where order does not matter

calculate the probability of an outcome or outcomes in situations where order does not matter

ex: how many ways are there to choose a 3 person committee from a class of 20?

ans: C(20,3) = 20! ÷ [ (20-3)! 3! ] = 1140

4.7 Combinations

ex: from a group of 5 men and 4 women, how many committees of 5 can be formed with a. exactly 3 women b. at least 3 women

ans a:

ans b:

404103

4

2

5

454

4

1

5

3

4

2

5

Combinatorics (§4.6 & 4.7)

Permutations – order matters E.g. President

Combinations – order does not matter E.g. Committee

Test Review

pp. 268-269 #1, 4, (5-6) ace…, 8, 9, 11 p. 270 #1, 2

Warm up A ferris wheel holds 12 riders. If there are 20 people waiting to ride it, how many ways can they ride it?

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