Answer to Question #419 in Statistics and Probability for Tony
Observations have shown that in some areas in September there are 12 rainy days in average. What is the probability that out of eight randomly taken days in this month three days will be rainy?
There are 30 days in September. Thus, 8 days can be selected in N = (C30)^8 different ways. There are 12 rainy days, thus, there are 18 not rainy days. We are interested in situation when there are 3 rainy and 5 not rainy days among 8 randomly taken days.
3 rainy days can be selected out of 12 rainy days in (C12)^3 ways. 5 clear days can be selected out of 18 clear days in (C18)^5 ways. The total number of ways to select 3 rainy and 5 clear days is: m = (C12)^3 *(C18)^5.
Thus, the probability of having exactly 3 rainy days in the selection is: P = m/N = ((C12)^3 *(C18)^5)/(C30)^8 ≈ 0.322.