# A simple classical statistics problem that I have trouble with

This question was from my CAs. Nothing has been changed.

A local weatherman forecasts that there is a 65% chance of rain on the coming Saturday and a 65% chance of rain on the Sunday immediately following that Saturday. What can you conclude from the above weather forecast?
Choose one of the following options.

A. It will certainly rain during the weekend (that is, it will rain on at least one of the two weekend days).

B. The chance of rain on Saturday and Sunday is 42.25%.

C. The weather forecast does not sound right, as P(rain on Saturday)+ P(rain on Sunday) > 100%.

D. The chance of rain on Saturday and Sunday is greater than 50%, but less than 85%.

E. The chance of rain on Saturday and Sunday is between 30% and 65%.

F. None of the above.

Chance of raining on Saturday = 0.65 Sunday = 0.65 Therefore, chance of raining on Saturday and Sunday is 0.65*0.65 = 0.4225

Doesn't this means there are two answers, B and E? Which is more correct?

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If B is correct, D can't be .4225 is not greater than 50% – Peter Flom Sep 14 '12 at 9:48
Sorry I meant E. I edited. This question is in the actual test and nothing have been changed. Really puzzling. – kingboonz Sep 14 '12 at 9:53
Oh, that makes more sense. I'd say B is more correct because it is more precise. – Peter Flom Sep 14 '12 at 9:56
The existence of temporal correlation in weather patterns suffices to make F the only reasonable reply. Even if we take this to be an abstract question (with "weather" as a metaphor merely to make it more palatable, and with a perfect determination of all probabilities involved), because the question provides no information about the independence or lack thereof of the Saturday and Sunday events, F is still the correct choice for the same reason. – whuber Sep 14 '12 at 15:54
@whuber I don't think it is a contradiction. I just think that my statement was incomplete. What I had in mind is that the weather forecaster has some sort of statistical model based on knowledge of meteorology. This model takes day to day effects into account and allows for a prediction interval on the forecast. I think I should have said that this interval would be constructed around a point estimate of the prediction that it would rain both on Saturday and Sunday rather than assert 0.4225 as the point estimate. – Michael Chernick Sep 14 '12 at 16:16

Considered only as a test problem, not as a real-world problem, the answer is fairly straightforward. We are told $P(A)=P(B)=0.65$, where $A$ is rain on Saturday and $B$ is rain on Sunday. Ruling out options A and C, the others are about $P(A \cap B)$.

We know that $P(A\cap B) = P(A) + P(B) - P(A\cup B)$. Since this is a multiple choice problem, we'll use intuition instead of algrebra to think about what $P(A \cup B)$ could be; we'll then use this to calculate what $P(A\cap B)$ could be.

Thinking of a simple Venn diagram, this is the combined area within either of two circles. One extreme is that the circles overlap; then the combined area is $0.65$. The other extreme is that they don't overlap at all; the combined would then be $0.65+0.65=1.3$. This is impossible, as the probability must be less than $1$, so we know that they must overlap, and that $1$ is the largest possibility.

Therefore $0.65 \leq P(A\cup B) \leq 1$. Substituting into our earlier equation, we get $0.65 + 0.65 - 0.65 = 0.65$ and $0.65 + 0.65 - 1 = 0.3$.

So $0.3 \leq P(A\cap B) \leq 0.65$: choice E.

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I think Peter was joking with his comment that F is correct but i contend in a serious vain that F is the right answer and of course the only correct answer. The point is that forecasts are statistical ESTIMATES of the probability and not the ACTUAL probability. The other answers assume that the forecaster really knows those probabilities rather than estimates from meteorological models. That is one reason why I would reject answer B. Another reason being that B assumes the weather on Sunday is independent of Saturday and this is not necessarily a good assumption. I think there could be positive correlation because storm systems can stay in an area for more than one day.

An acceptable answer for me other than F would be a prediciton interval around the appropriate point estimate (not 42.25%) based on the uncertainty from the forecasters model.

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