I'm a complete newbie here, so forgive me if this seems trivial to any of you, but the proposed solution to the quiz seems wrong to me. In short: It is stated that the probability of no rain in the next 3 days is 90%. The objective is to figure out the probability of rain tomorrow. The proposed solution is "Cannot be determined".
I did the following calculations: Let p(N) be the probability of no rain on a particular day. The complement is p(R) (rain on a particular day).
If the probability of no rain on 3 consecutive days is 0.9, then p(N)^3 = 0.9. <--> p(N) = 0.9^(1/3) = 0.96549 p(R) = 1 - 0.96549 = 0.03451
The only combinations of rainy vs non-rainy days with rain tomorrow are R-R-R, R-R-N, R-N-R, and R-N-N. Since we know the probabilities of rain and no rain on a particular day, we can calculate the probability of rain tomorrow:
p(rain tomorrow) = p(R-R-R) + p(R-R-N) + p(R-N-R) + p(R-N-N) = p(R)^3 + 2(p(R)^2 * p(N)) + p(R)p(N)^2 = 0.96549^3 + 2*(0.96549^2 * 0.03451) + (0.96549 * 0.03451^2) = 0.0345
So the probability of rain tomorrow should be 3.45%, or the same as the probability of rain on any particular day.
So, either I made a mistake or this calculation is invalid or the proposed answer that there is not enough information given to solve the problem is wrong (maybe what was meant is that with the information given in the pages leading up to this problem we couldn't solve this problem?). Can anyone confirm this or show me what I did wrong?