Independence Testing with Two Events

6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly’s restaurant.
\(S =\) the event a customer has a starter.
\(M =\) the event a customer has a main course.
\(D =\) the event a customer has a dessert.
\includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events \(S\) and \(D\) are statistically independent

1. find the value of \(p\).
2. Hence find the value of \(q\).
3. Find
   1. \(\quad\) P( \(D \mid M \cap S\) )
   2. \(\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)\) One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
4. Estimate the number of desserts that these 63 customers will have.

1 For the events \(A\) and \(B\) it is given that $$\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.3 \text { and } \mathrm { P } ( A \text { or } B \text { but not both } ) = 0.4 \text {. }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\).
3. State, giving a reason, whether \(A\) and \(B\) are independent.

4. A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5 , the probability of winning the second is 0.3 and the probability of winning both projects is 0.2 .

1. Find the probability that he does not win either project.
2. Find the probability that he wins exactly one project.
3. Given that he does not win the first project, find the probability that he wins the second.
4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events.