WJEC Maths for A2 – Applied

1 Probability 16 Answer 3 (a) There are two ways in which the defendant could be found guilty. He/she could have committed the crime and be found guilty. He/she could have not committed the crime and be found guilty. Let C = event they committed the crime G = event they were found guilty C C Ԣ P( C ) = 0.8 P( C Ԣ ) = 0.2 P( G | C ) = 0.9 P( G Ԣ | C ) = 0.1 P( G | C Ԣ ) = 0.05 P( G Ԣ | C Ԣ ) = 0.95 G G Ԣ G G Ԣ Hence, probability found guilty = P( C ) × P( G | C ) + P( C′  ) × P( G | C′  ) = 0.8 × 0.9 + 0.2 × 0.05 = 0.73 (b) Pȍ C ∩ G  Ȏ = Pȍ C  Ȏ Pȍ C  | G  Ȏ Pȍ C  | G  Ȏ = Pȍ C ∩ G  Ȏ Pȍ C  Ȏ = 0.8 × 0.9 0.73 = 72 73 Step by A group of university students took part in a survey about student lifestyles. Of these students, 60% claimed to exercise regularly. The probability of a student claiming to exercise regularly and telling the truth is 0.32. The probability of a student claiming to not exercise regularly and telling the truth is 0.97. (a) Using a tree diagram, or otherwise, show that the probability of a person who does not exercise regularly also claiming they don’t exercise regularly is 0.388. A randomly selected student completes the survey. (b) Find the probability that the student is telling the truth. (c) Given that the student is telling the truth, ϐind the probability that the student claims to exercise regularly. Steps to take 1 You need to understand the question before you start. When you read this question carefully you can see there are two events: claiming to exercise regularly or not and the claim being true or false. A tree diagram is drawn. Notice that the ϐirst event is whether the person committed the crime or not and the second event is whether they were subsequently found guilty or not. There are two paths that will ϐind the person guilty. The probability of each path is found and the required probability is the sumof these. Note that Pȍ C ∩ G  Ȏ is the probability that they committed the crime and are found guilty.