independence and tree diagrams slideshow 56, mathematics mr richard sasaki, room 307

Independence and Tree Diagrams

Slideshow 56, Mathematics

Mr Richard Sasaki, Room 307

Objectives

• Review multiplying decimal numbers less than 1

• Learn some new notation about different events

• Recall how to make calculations with independent events

• Introduce Tree Diagrams

Decimal Numbers

We will start using decimal numbers in probability (as well as fractions).

Let’s have a bit of practice multiplying and dividing some!

0.03 0.18

0.21 0.42

0.021 0.092

0.0045 0.07

0.5 0.3

0.9 0.3

0.125 0.08

0.01 0.375

Notation

Do you remember how to calculate the probability of two independent events occurring successfully?ExampleWe roll an unbiased regular die and a spin an unbiased spinner with values 1, 2 and 3 in that order.

What is the probability of getting a 5 and then a 2?

P(5, 2) = 16×13

¿118

Notation

When independent events take place, we multiply both of their probabilities to find the probability of them both occurring.

P(AB) = P(A) P(B)Note – This is only true when events A and B are independent!Also, the complement of A is written A’.

P(A’) P(A)

Do you remember how to write “the probability of event A and B”?

P(AB)

P(A’) = 0.6

P() = 0.4 )

P() = 0.4 )

P() = 0 because A can’t happen and not happen at the same time.)

Replacing / with replacement)

P(A) = P(Jack) =

P(B) = P(Red) =

P(A’) = 1 – P(A) =

P() )

BecauseP(B) = P(B’) = ½

Tree Diagrams

We can represent different possible outcomes with tree diagrams. Branches represent independent events. Let’s consider two events with only success and fail outcomes.

Event A Event B

P(A)

P(A’)

P(B)P(B’)

P(B)P(B’)

P(AB)

P(AB’)P(A’B)

P(A’B’)

Example

Event A Event B

P(A)

P(A’)

P(B)

P(B’)

P(B)

P(B’)

P(AB)

P(AB’)P(A’B)

P(A’B’)

Two independent events A and B occur in order where P(A) = 0.3 and P(B) = 0.9. Represent all outcomes with a tree diagram and calculate P(AB’) and P(A’B’).

0.3

0.7

0.9

0.9

0.1

0.1

=0.03

=0.07

Answers - Easy

P()

P()

P()

P()

P()P()

12

12

1212

1212

P() )P() )

P() P()

P()

P()

P()

P()

P()

P()

0.25

0.75

0.2

0.20.8

0.8

P() )

P() )

P(AB) = P(A) P(B).BecauseSo A and B must be independent.

Answers - Hard)

)

))

A B

1747

27

27

27

27

37

37

37

27

27

27

1000)

100)

P(Other)

1000)

1000)

1000)

100)

100)

100)

P(Other)

P(Other)

)

P(Two ￥ 100) =

P(￥ 1000 and ￥ 100) =

P(At least one ￥ 1000) =

P(No ￥ 100 or ￥ 1000) =