An electronics shop has an alarm at the door that is triggered when a customer attempts to leave the shop with stolen goods.

However, the alarm can also be triggered falsely due to other items the customer is carrying or wearing. Whenever the alarm is triggered, the customer is searched. A record is made each time there is a genuine attempted theft.

The manager wishes to investigate the accuracy of the alarm.

The shop has a counter on the door, so she knows how ma customers have exited the shop in any period of time. There is also a counter on the alarm that records the total number of triggers. After consulting records from the past month, the manager finds that • A total of 1960 customers exited the shop.
• The alarm was triggered 68 times.
• Five of the trigger events were attempted thefts.
Define the sample space Ω = {all occasions on which customers exit the shop}, and events A = {alarm is triggered}, T = {attempted theft}.
You may assume that all attempted thefts are successfully detected by the alarm.
Specifying a model means defining a random variable and proposing a suitable distribution for it, which should involve the unknown parameter you need to estimate.

a. For each of the following probabilities, say whether its value is known or whether it needs to be estimated from the sample data. If it needs to be estimated, suggest a suitable model and write down its maximum likelihood estimate under that model. Specify any assumptions you make in your model formulations.
i. P(A)
ii. P(T)
iii. P(A | T)
iv. P(T | A)

b. Which of the probabilities in part

(a) should the manager use for her investigation of the accuracy of the alarm?

c. When the alarm goes off again in future, what do you estimate is the probability that it is a false alarm?