Bayes formula (a) In a day a store serves with equal probability a number of cus

Question

Bayes formula (a) In a day a store serves with equal probability a number of customers which is between zero and five (including zero and five). What is the probability that there will be no customers during at least one of two consecutive days? (b) You are having an outdoor wedding ceremony in a desert tomorrow. The weather forecast pre dicts a rainy day. You do know that historically, the forecast accurately predicted rain 90% of time when it does indeed rain. When it doesn't rain, it incorrectly predicts rain 10% of the time It also does not rain very often in this location: usually only 5 rainy days are observed during the year and you are not aware of any seasonal or other pattern of distribution of these days. What is the probability that it will rain tomorrow, and you will have to alter your wedding plans? (c) A company estimates that 0.5% of its products have defects. A customer who buys a defective product, returns it with probability 60%, 5% of customers who's product is not defective will return it as well. Find the probability that a returned product is defective

Explanation / Answer

(a)
probability that there will be no customer, p = 1/6
probability that there will be no customers during at least one of two consecutive days = 2C1*p*(1-p) + p^2

Probability = 2*(1/6)*(5/6) + (1/6)^2 = 0.3056

(b)
probability of rain, P(R) = 5/365 = 0.0137
P(R') = 1 - 0.0137 = 0.9863

Probability that it will rain when predicted, P(P|R) = 0.9
P(P|R') = 0.1

Required Probability, P(R|P) = P(P|R)*P(R)/(P(P|R)*P(R) + P(P|R')*P(R')) = 0.9*0.0137 / (0.9*0.0137 + 0.1*0.9863) = 0.1111

(c)
P(D) = 0.005
P(R|D) = 0.6 and P(R|D') = 0.5

P(D|R) = P(R|D)*P(D) / (P(R|D)*P(D) + P(R|D')*P(D'))
P(D|R) = 0.6*0.005 / (0.6*0.005 + 0.5*0.995) = 0.006

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