# If today is sunny, what is the probability that day after tomorrow will be cloudy?

I came across this problem and suspect that the language of the problem is faulty. Does one assume the probability of it being sunny the first day 0.7 ? Are there three days in this problem? What would a tree diagram look for this problem?

My working for part a) S-S-C+ S-C-C (0.7*0.7*0.3 + 0.7*0.3*0.6)

b) S-S-C + S-C-S (0.7*0.7*0.3 + 0.7*0.3*0.4)

Here is the problem:

Meteorological data shows that if today is sunny weather, then probability that tomorrow also will be sunny is 0.7, and in case if today is cloudy, then tomorrow will be sunny with a 0.4 probability.

a) If today is sunny, find the probability that day after tomorrow will be cloudy

b) If today is sunny, find the probability that only one of the next two days will be sunny

• The language of the problem is fine. The wording "If today is sunny..." means the same as "Given that today is sunny,...". So you don't need those 0.7's. – TonyK Apr 2 '17 at 11:33
• What would the working for this problem? And the final answer? – pirsquare Apr 2 '17 at 11:38
• Doesn't "day after tomorrow " imply 3 days? – pirsquare Apr 2 '17 at 11:45

Let today, tomorrow and the day after tomorrow correspond with $0,1,2$ respectively.

Then in your work on a) you actually calculate $\Pr(S_0\cap C_2)$ under your own (hence questionable) assumption that $\Pr(S_0)=0.7$.

However, you are asked to calculate $\Pr(C_2\mid S_0)$.

You make the same mistake in b).

Dividing both original answers by $\Pr(S_0)$ will repair.

This comes to the same as striping away the first 0.7 (as TonyK) suggests in his comment.

• I got it now. So my the working for Part a) should be [(0.7*0.7*0.3 + 0.7*0.3*0.6)]/0.7 = 0.39 – pirsquare Apr 2 '17 at 12:03
• That's a way to do it, but the shorter $0.7\times0.3+0.3\times0.6=0.39$ is more elegant. – drhab Apr 2 '17 at 12:05
• Yes, the 0.7 from the denominator cancels out the ones in the numerator. – pirsquare Apr 2 '17 at 12:07
• It is not only a notational question. You come to this more direct solution by making use of the data: "today it is sunny". – drhab Apr 2 '17 at 12:10
• Are you suggesting the key insight here is that the numerical value of S0 is irrelevant or it may not even be 0.7? – pirsquare Apr 2 '17 at 12:22

No, the language of the question is not faulty. What you have to evaluate is the conditional probability that given today is sunny, find out the probability that day after tomorrow is cloudy.
What you have calculated is the probability of day after tomorrow being cloudy, assuming that the probability that it is sunny today being 0.7. Of course this probability would be lower as the sample space includes the event that today is cloudy. Thus, the correct calculations should be:$$a)\ P = S-C + C-C$$ The next problem follows similarly. Note: S-C denotes the probability of tomorrow being sunny and day after tomorrow being cloudy as declared in the question details.

• MathJaxing S-C + C-C has turned it into "S minus C plus C minus C", which is rather confusing! And in any case I suggest the notation (S)-S-C + (S)-C-C. – TonyK Apr 2 '17 at 11:42
• @TonyK Hope it's clear now. (S)-S-C would again imply that we have to consider P(S), so I didn't use it. – bat_of_doom Apr 2 '17 at 13:44
• It's the spacing that makes it unintelligible. – TonyK Apr 2 '17 at 14:12