Probability : Combined events 2

Objectives

When you have competed it you should

* know the conditional probability

* know the independent events

Key terms: Independent events, With/without replacement, ‘Given’ situation

* know the multiplication rule

Example 1Consider a class of thirty students, of whom sixteen are girls and fourteen are boys.

Suppose further that four girls and five boys are left-handed, and all remaining students are right-handed.If a student is selected at random from the whole class then the chance that he or she is left handed is

30

9)( handedleftP = 3/10

However , suppose now that student is selected at random from the boys in the class. The chance that the boy will be left-handed is

14

5

This is an example of conditional probability

Formulae for conditional probability If A and B are two events and P(A) > 0, then the conditional probability of B given A is

)(

) and ()A (

AP

BAPBP

Rearranging this equation gives P(A and B) = P(A) × P(B | A)

This is known as the multiplication law of probability.

Example 2

Events A and B are such that P(A) = 0.6 , P(B) = 0.7 and P(AB) = 0.4.

(i)

Solutiom

(i) Find P(A B)

A B1 = 0.2 A1 B = 0.3A B = 0.4

(ii) Find P(B A)

P(AB) = 0.4/0.7 = 4/7

(ii) P(BA) = 0.2/0.6 = 1/3

Independent events

If two events A and B are independent then

P(A and B) = P(A) x P(B)

i.e. P(BA) = P(B) or P(AB) = P(A)

Independent events are events which have no effects on one another.

Example 3

A fair die is thrown twice. Find the probability that two threes are thrown.

Solution: Independent events

P(3 and 3) = 1/6 x 1/6 = = 1/36

Tree diagrams

ExampleA bag contains 4 red and 5 green counters. A counter is drawn at random and not replaced. A second counter is then selected.

Find the probability of selecting two green counters.Tree diagram

4/9

5/9

3/8

5/8

4/8

4/8 P(G and G) =

5/9 x 4/8 = 20/72 = 5/18

Tree diagramsExample 1In a class of 24 girls, 7 have black hair. If 2 girls are chosen at random from the class, find the probability that (i) both have black hair (ii) neither have black hair

Tree diagram

6/23 B B and B 7/92

B

7/24

17/23 NB B and NB 119/552

7/23 B NB and B 119/55217/24

NB

16/23 NB NB and NB 34/69

Check 1