Question

Today is Sunday. Based on the weather forecast, the probability of rain on Monday, Tuesday, and Wednesday is 20%, 10%, and 60%, respectively. The probability of rain on both Monday and Tuesday is 5%. If it rains on Tuesday, the probability of rain on Wednesday is 75%.

a. Are the events “rain on Monday” and “rain on Tuesday” independent? Explain.

b. Are the events “rain on Tuesday” and “rain on Wednesday” independent? Explain.

Answer 1

We have here:
P( monday rain) = 0.2,
P( tuesday rain) = 0.1,
P( wednesday rain) = 0.6

Also, we are given here that:
P( monday and tuesday rains) = 0.05,
P( wednesday rain | tuesday rain) = 0.75

a) Using Bayes theorem, we have here:
P( monday rain | tuesday rain) = P( monday and tuesday rains) / P( tuesday rain)

= 0.05 / 0.1 = 0.5 which is not equal to P( monday rain)

Therefore the events rain on Monday and rain on tuesday are not independent events.

This is because for independent events,
P(X | Y) = P(X)

b) Again using the same method as in above part, we have here:
P( wednesday rain | tuesday rain) = 0.75 which is not equal to P( wednesday rain) = 0.6

Therefore the events rain on wednesday and rain on tuesday are not independent events.