WJEC Maths for A2 – Applied

1.2 The conditional probability formula 15 If the values for P( A ∩ B ) obtained using both formulae are the same, then events A and B must be independent. The following examples show this method. Examples 1 If two events A and B occur such that P( A ) = 0.4 and P( B ) = 0.8 and P( A  ∩  B ) = 0.32, determine whether or not these two events are independent. Answer 1 If the events are independent, then P( A ∩ B ) = P( A ) × P( B ). We can enter the values for the probabilities of A and B respectively and then compare the value of P( A ∩ B ) calculated with the value in the question. If they are the same, then A and B are independent events. P( A ∩ B) = P( A ) × P( B ) = 0.4 × 0.8 = 0.32 This value is the same as the value in the question which means A and B are independent events. 2 Two events A and B are such that P( A ) = 0.2, P( B ) = 0.6 and P( A ∪ B ) = 0.4. Decide whether events A and B are dependent or independent events. Answer 2 The generalised addition law that links the probability of the intersection (A ∩ B) with the probability of the union ( A ∪ B ) can be used here. Pȍ A ∩ B Ȏ = Pȍ A Ȏ + Pȍ B Ȏ − Pȍ A ∪ B Ȏ = 0.2 + 0.6 − 0.4 = 0.4 If the events A and B are independent, the probability of both A and B occurring is found by multiplying P( A ) and P( B ) together. So if the events are independent Pȍ A ∩ B Ȏ = P( A ) × P( B ) = 0.2 × 0.6 = 0.12 Now 0.4 ≠ 0.12 so the events A and B are not independent. A and B are therefore dependent events. 3 In a certain country, 80% of the defendants being tried in the Law Courts actually committed the crime. For those who committed the crime, the probability of being found guilty is 0.9. For those who did not commit the crime, the probability of being found guilty is 0.05. (a) Find the probability that a randomly chosen defendant is found guilty. (b) Given that a randomly chosen defendant is found guilty, ϐind the probability that this defendant committed the crime. P( A ∩ B ) is the probability of A and B occurring. The generalised addition law is included in the formula booklet. Note if the events A and B were independent events, the two probabilities worked out would have been equal.