Just do (e). Thanks

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.

1. (a)Show that the probability that Peter will leave an umbrella in the restaurant on a certain day is 0.135. [4]
2. (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]
3. (c)Determine whether the following two events are indepedent or not: [3]E1: It will be raining at 7 pm tomorrow.
4. E2: Peter will leave an umbrella in the restaurant tomorrow.
5. (d)Find the probability that Peter will leave more than one umbrella in the restaurant in a period of 5 days. [5]
6. (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. 

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 pm7 he will certainly bring an umbrella with him. If it is not raining7 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 denitely 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] 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 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] 00 . 1 Hint: You ma use the identit irhl = for r