At a certain fishing spot two consecutive rainy day are twice as likely to be fo

Question

At a certain fishing spot two consecutive rainy day are twice as likely to be followed by another rainy day than by a sunny day, while two consecutive sunny day are five times more likely to be followed by another sunny day. It one of two consecutive days is rainy and the other sunny, they are equally like to be sunny or rainy. A fisherman observes that the number of catches per day is Poisson distributed with mean 4 in rainy days and mean 1 in sunny days. He visits the spot on Saturday and Sunday, which are both sunny day. (a) Find the probability that Tuesday is a sunny day. (b) Find the mean number of fish expected for Tuesday.

Explanation / Answer

Saturday - Sunny, Sunday - Sunny

Monday- probability of sunny day = 5/6; probability of rainy day = 1/6.

Tuesday:

Probability of sunny day = probability of sunny day, given monday was sunny + probability of sunny day, given monday was rainy = 5/6 x 5/6 + 1/6 x 1/2 = 25/36+1/12=28/36

(b) probability of Tuesday being rainy day = probability of rainy day, given monday was sunny+ probability of rainy, given monday was rainy = 5/6 x 1/6 + 1/6 x 1/2= 8/36.

Expected number of catch n Tuesday = (8/36)x4+ 28/36x1= 60/36 = 5/3 = 1.67

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