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Peter goes out at 7 pm for dinner in a restaurant and leaves the restaurant at 8 pm everyday. If it is raining at 7 pm7 he will certainly bring an...
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7. Peter goes out at 7 pm for dinner in a restaurant and leaves the restaurant at 8 pm everyday. If it is raining at 7 pm, he will certainly bring an umbrella with him. If it is not raining, he will do so with probability 0.3. If it is not raining at 8 pm, he has 0.5 probability to leave his umbrella in the restaurant. If it is raining at 8 pm, he will definitely not leave any umbrella in the restaurant. Suppose that the weather on a certain day is independent of that on any other days. Past experience reveals that: [20 marks]
• There is 0.2 probability to be raining at 7 pm on a certain day.
• If it is raining at 7 pm on a certain day, there is 0.73 probability that it will be raining at 8 pm.
• If it is not raining at 7 pm on a certain day, there is 0.1 probability that it will be raining at 8 pm.
- (a) Show that the probability that Peter will leave an umbrella in the restaurant on a certain day is 0.135. [4]
- (b) Given that Peter left an umbrella in the restaurant on a certain day, what is the probability that it was raining at 7 pm on that day? [2]
- (c) Determine whether the following two events are indepedent or not: [3]E1 : It will be raining at 7 pm tomorrow.
- E2 : Peter will leave an umbrella in the restaurant tomorrow.
- (d) Find the probability that Peter will leave more than one umbrella in the restaurant in a period of 5 days. [5]
- (e) Peter left an umbrella in the restaurant tonight. After how many days, on the average, will Peter leave another umbrella in the restaurant? [6]
- 7. Peter goes out at 7 pm for dinner in a restaurant and leaves the restaurant at 8 pm everyday. If it is raining at 7pm7 he will certainly bring an umbrella with him. If it is not raining7 he will do so with probability 0.3. If it isnot raining at 8 pm, he has 0.5 probability to leave his umbrella in the restaurant. If it is raining at 8 pm, he willdefinitely not leave any umbrella in the restaurant. Suppose that the weather on a certain day is independent ofthat on any other days. Past experience reveals that: [20 marks] 0 There is 0.2 probability to be raining at 7 pm on a certain day. 0 If it is raining at 7 pm on a certain day, there is 0.73 probability that it will be raining at 8 pm.0 If it is not raining at 7 pm on a certain day, there is 0.1 probability that it will be raining at 8 pm. (a) Show that the probability that Peter will leave an umbrella in the restaurant on a certain day is 0.135. [4] (b) Given that Peter left an umbrella in the restaurant on a certain day7 what is the probability that it was rainingat 7 pm on that day? [2] (c) Determine whether the following two events are indepedent or not: [3] E1 : It will be raining at 7 pm tomorrow.E2 : Peter will leave an umbrella in the restaurant tomorrow. (d) Find the probability that Peter will leave more than one umbrella in the restaurant in a period of 5 days. [5] (e) Peter left an umbrella in the restaurant tonight. After how many days, on the average, will Peter leave anotherumbrella in the restaurant? [6] 00. 1 Hint: You ma use the identit irhl = — for r < 1. [ y y 2: (1 _ T). I | 1 1:1
