1. Probability : Combined events 2
Objectives
When you have competed it you should
* know the multiplication rule
* know the conditional probability
* know the independent events
Key terms: Independent events, With/without replacement, ‘Given’ situation

2. Example 1
Consider 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
= 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
This is an example of conditional probability

3. Formulae for conditional probability
If A and B are two events and P(A) > 0, then the conditional probability of B given A is
Rearranging this equation gives
P(A and B) = P(A) × P(B | A)
This is known as the multiplication law of probability.

4. Example 2
Events A and B are such that P(A) = 0.6 , P(B) = 0.7 and P(AB) = 0.4.
(i) Find P(A B)
(ii) Find P(B A)
Solutiom
A  B1 = 0.2
A1  B = 0.3
A  B = 0.4
(i)
P(AB) = 0.4/0.7 = 4/7
(ii)
P(BA) = 0.2/0.6 = 1/3

5. Independent events
Independent events are events which have no effects on one another.
If two events A and B are independent then
P(A and B) = P(A) x P(B)
i.e. P(BA) = P(B) or P(AB) = P(A)
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

6. Tree diagrams
Example
A 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
3/8
5/8
4/9
4/8
5/9
4/8 
P(G and G) =
5/9 x 4/8 =
20/72 =
5/18

7. Tree diagrams Example 1 In 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