Finding Probability Using Tree Diagrams and Outcome Tables

[Pages:40]Finding Probability Using Tree Diagrams and Outcome Tables

Chapter 4.5 ? Introduction to Probability

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Tree Diagrams

n if you flip a coin twice, you can model the possible outcomes using a tree diagram or an outcome table resulting in 4 possible outcomes

H H

T

H T

T

Flip 1

H H T T

Flip 2

H T H T

Simple Event

HH HT TH TT

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Tree Diagrams Continued

n if you rolled 1 die and then flipped a coin you

have 12 possible outcomes

H

1

T

(1,H) (1,T)

2

H

(2,H)

T

(2,T)

H

3

T

(3,H) (3,T)

H

4

T

(4,H) (4,T)

H

(5,H)

5

T

(5,T)

H

(6,H)

6

T

(6,T)

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Sample Space

n the sample space for the last experiment would be all the ordered pairs in the form (d,c), where d represents the roll of a die and c represents the flip of a coin

n clearly there are 12 possible outcomes (6 x 2) n there are 3 possible outcomes for an odd die

and a head n so the probability is 3 in 12 or ? n P(odd roll, head) = ?

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Multiplicative Principle for Counting

n The total number of outcomes is the product of the possible outcomes at each step in the sequence

n if a is selected from A, and b selected from B... n n (a,b) = n(A) x n(B)

q (this assumes that each outcome has no influence on the next outcome)

n How many possible three letter words are there?

q you can choose 26 letters for each of the three positions, so there are 26 x 26 x 26 = 17576

n how many possible license plates are there in Ontario?

q 26 x 26 x 26 x 26 x 10 x 10 x 10

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Independent and Dependent Events

n two events are independent of each other if an occurence in one event does not change the probability of an occurrence in the other

n what is the probability of getting heads when you know in advance that you will throw an even die?

q these are independent events, so knowing the outcome of

the second does not change the probability of the first

P (heads

| even ) =

P (heads

even )

=

3 12

=

1

P (even )

3 6

2

as P (heads ) = 1 , we can say P (heads | even ) = P (heads ) 2

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Multiplicative Principle for Probability of Independent Events

n we know that if A and B are independent events, then...

q P(B | A) = P(B) q if this is not true, then the events are dependent

n we can also prove that if two events are independent the probability of both occurring is...

q P(A and B) = P(A) ? P(B)

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Example

n a sock drawer has a red, a green and a blue sock n you pull out one sock, replace it and pull another out n draw a tree diagram representing the possible outcomes n what is the probability of drawing 2 red socks? n these are independent events

R P(red and red )

R

B

G = P(red ) ? P(red )

R

B

B G

= 1?1 = 1

R

33 9

G

B

G

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