WJEC Maths for A2 – Applied

1 Probability 10 1.1 Using tree diagrams for conditional probability You came across tree diagrams in your GCSE studies. Tree diagrams are diagrams used to represent the probabilities of combined events. Each path (which is a branch of the tree) corresponds to a certain sequence of events. By multiplying the probabilities of the separate events along the path you can work out the probability of a particular sequence of events. If the required probability involves several paths, the probabilities of each path are found and then added together to give the required probability. Tree diagrams for conditional probability Conditional probability is the probability of an event occurring given that another event has occurred. Suppose you have a bag containing 6 red and 4 black balls and one ball is picked at random and its colour noted and kept out of the bag. Another ball is picked at random and its colour noted. We can draw the following tree diagram to show the probabilities: Notice that on the second pick the probability has changed depending on what colour ball was removed ‹ –Š‡ ϐ‹”•– ’‹…Ǥ Red Black P ( Red ) = 10 P (Black) = 10 6 4 P ( Red ) = 9 P (Black) = 9 5 4 P ( Red ) = 9 P (Black) = 9 6 3 Red Black Red Black If the ϐirst ball had been put back into the bag ready for picking the second ball, the probabilities would have been the same as the ϐirst pick. In the situation shown by the tree diagram, the probability of obtaining each colour has changed depending on what colour ball was removed in the ϐirst pick. Hence, this is conditional probability. For example, if you wanted the probability that the ϐirst ball was black and the second ball was red you multiply the probabilities along the path like this P(black and then red) = P(black) × P(red given ϐirst ball was black) We can write this using the following shorthand notation P( B ∩ R ) = P( B ) × P( R | B ) = 4 10 × 6 9 = 4 15 Suppose there are two events A and B . The probability of B occurring given that A has already occurred is written as P( B | A ). For short, we can say that P( B | A ) is the probability of B given A . Note that P( B ∩ R ) is the probability of obtaining a black and then a red ball, P( R | B ), is the probability of obtaining a red ball given that a black had been obtained from the ϐirst pick.