Conditional probability Objectives When you have competed it you should * know the multiplication law of conditional probability, and be able to use tree diagrams * know the multiplication law for independent events

Example 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

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.

Tree diagrams Example 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

Tree diagrams Example 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 (i) (ii)

Example A company is worried about the high turnover of its employees and decides to investigate whether they are more likely to stay if they are given training. On 1 January Neither year the company employed 256 people. during that year a record was kept who received training as well as who left the company. Still employed Left company Given training Not given training

Still employed Left company Given training Not given training Find the probability that a randomly selected employee (a) received training(b) did not leave the company (c) received training and did not leave the company (d) did not leave the company, given that he person had received training did not leave the company, given that the person had not received training

Solution T: The employ received training S: The employee stayed in the company (a) P( T ) = 152 / 256 (b) P( S ) = 169 / 256 (c) P(T and S) = 109 / 256 (d)= 109 / 152 (e)= 60 / 104 Exercise 4B