Tree Diagrams

Slideshow 58, MathematicsMr Richard Sasaki, Room 307

Objectives

• Review how tree diagrams appear• Draw tree diagrams for three events• Introduce tree diagrams where events are

dependent

Tree Diagrams

We know that tree diagrams can hold information for over two events with success / 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’)

Tree Diagrams

Sometimes however, there are more branches, or even three events.

Event AEvent B

P(A)

P(A’)

P(B)

P(B’)

P(B)

P(B’)

Event CP(C)

P(C)

P(C)

P(C)

P(C’)

P(C’)

P(C’)

P(C’)

P(ABC)P(ABC’)P(AB’C)

P(AB’C’)P(A’BC)

P(ABC’)P(AB’C)

P(AB’C’)

We multiply three numbers for each branch instead of two. It’s the same idea!

Answers - Easy

Coin A Coin BCoin C

P(A)

P(A’)

P(B)

P(B’)

P(B)

P(B’)

P(C)P(C’)P(C)P(C’)P(C)P(C’)P(C)P(C’)

=

= =

= =

= =

=

𝑃 (𝐴∩𝐵∩𝐶)𝑃 (𝐴∩𝐵∩𝐶 ′ )𝑃 (𝐴∩𝐵 ′ ∩𝐶 )𝑃 (𝐴∩𝐵 ′ ∩𝐶 ′)𝑃 (𝐴 ′ ∩𝐵∩𝐶 )𝑃 (𝐴 ′ ∩𝐵∩𝐶 ′)𝑃 (𝐴 ′ ∩𝐵 ′ ∩𝐶 )𝑃 (𝐴 ′ ∩𝐵 ′ ∩𝐶 ′)

Event A Event BEvent C

P(A)

P(A’)

P(B)

P(B’)

P(B)

P(B’)

P(C)P(C’)P(C)P(C’)P(C)P(C’)P(C)P(C’)

12

12

1212

12

12

1212121212121212

13

13

13

23

23

23

13

13

13

13

23

23

23

23

P(A)=

P(A)=

P()= or

P(A’)=

P(1 at least twice)=

1 - P(A’)=

Answers - Hard

𝑃 (𝐴∩𝐵1)𝑃 (𝐴∩𝐵1 ′ )𝑃 (𝐴 ′ ∩𝐵2)𝑃 (𝐴 ′ ∩𝐵2 ′)

No. The outcome of A clearly changes the probability of B being successful.

Event A Event BEvent C

P(A)

P(A’)

P(B)

P(B’)

P(B)

P(B’)

P(C)P(C’)P(C)P(C’)P(C)P(C’)P(C)P(C’)

0.30.5

0.50.7

0.5

0.5

0.1

0.1

0.1

0.1

0.9

0.9

0.9

0.9

0.7 0.015

0.315 0.15

0.05

Dependence

What is dependence?Dependence is the opposite of independence. It means that two (or more) things do affect each others outcome.

Event A Event B

P(A))

P(A’)

0.4

0.6

0.9

0.10.2

0.8

In this example, we can see if A is successful, B is likely to be successful too. But if A fails, B is likely to fail as well.

Conditional Probability

Now for dependent events A and B, on a question on the last worksheets, we saw B1 and B2. This is because we knew .

Event A Event B

P(A)

P(A’)

0.4

0.6

0.9

0.10.2

0.8

𝑃 (𝐵1)

𝑃 (𝐵1 ′)𝑃 (𝐵2)

𝑃 (𝐵2 ′)

What makes B1 and B2 different?

A causes them to be. (they are dependent of A.)

Conditional Probability

What are the meanings of though? How can we define between them?

Event A Event B

P(A)

P(A’)

0.4

0.6

0.9

0.10.2

0.8

𝐵1

𝐵2

B1 is B .given that A was successfulB2 is B .given that A was unsuccessful

If something happens differently because of something else, we call this a condition. This is why this is conditional probability.

Notation – Conditional Probability

“Given that” is denoted by the symbol ‘ ’.

Event A Event B

P(A)

P(A’)

|So the probability of B being successful given that A was successful is denoted as .P(B|A)

P(B|A)P(B’|A)

P(B|A’)

P(B’|A’)

𝑃 (𝐴∩𝐵)𝑃 (𝐴∩𝐵 ′ )𝑃 (𝐴′∩𝐵)

𝑃 (𝐴 ′ ∩𝐵 ′)

= 𝑃 (𝐴)∙𝑃 (𝐵∨𝐴)

ExampleA forgetful man leaves his house. There is chance that he remembers to lock his door. If he locks his door, there is a chance of him being robbed. But if he forgets, the chance is .

Event A Event B

P(A)

P(A’)

P(B|A)P(B’|A)

P(B|A’)

P(B’|A’)

𝑃 (𝐴∩𝐵)𝑃 (𝐴∩𝐵 ′ )𝑃 (𝐴′∩𝐵)

𝑃 (𝐴 ′ ∩𝐵 ′)

910

110

12000199920001300

299300

=

=

=

=

Event A Event B

P(A)

P(A’)

P(B|A)

P(B’|A)

P(B|A’)

P(B’|A’)

0.7

0.3

0.90.1

0.30.7

0.630.90.3

0.09

Event A Event B

P(A)

P(A’)

P(B|A)

P(B’|A)

P(B|A’)

P(B’|A’)

7002000

13002000

1201920

34

14

P(B|A) =

𝑃 ( 𝐴∩𝐵 )= 70040000

Event A Event B

P(A)

P(A’)

P(B|A)

P(B’|A)

P(B|A’)

P(B’|A’)

0.95

0.05

0.70.3

0.350.65

0.285A

0.050.3

1 - 1 – 0.0325 = 0.9675

Event A Event B

P(A)

P(A’)

P(B|A)

P(B’|A)

P(B|A’)

P(B’|A’)

3050

2050

210009981000

250

4850

9602500𝐵𝐴250