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I have this problem:

Probability that it rains during one day is $0.4$ and probability that is does not is $0.6$. a) What is a distribution of $X$ - number of rainy days after the day without rain? b) What is a probability that we have two consecutive rainy days?

Is this the way how to solve a):

If there is one rainy day then probability is: $P = \frac{0,4 \times 0,6}{0,6}$

If there are two rainy days then probability is: $P = \frac{0,4 \times 0,6^{2}}{0,6}$, etc...

Thank you!

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  • $\begingroup$ to b) : Do you mean: what is the probability, that e.g. tomorrow and the day after tomorrow (i.e. two specific days) are rainy? Or do you mean like: the probability that we have two consecutive rainy days in a period of time (like a week)? $\endgroup$ – Bernd Jan 16 '14 at 8:23
  • $\begingroup$ @Bernd That is also what is confusing me. Example is written exactly like that. $\endgroup$ – Novak Djokovic Jan 16 '14 at 10:18
  • $\begingroup$ Yes, after reading it again and reading André's answer, I think he is right. Any not-rainy day can be considered at start, and then X = 2 means RRN. Probabilities see in his very good answer. $\endgroup$ – Bernd Jan 16 '14 at 11:09
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We assume independence, which is very unreasonable. But without some assumption we cannot solve the problem.

The random variable $X$ is the number of consecutive rainy days after a rainless day. So $X$ takes on values $0,1,2,3,\dots$.

The probability that $X=0$ is $0.6$.

For $X=1$, we need RN (rain, then not rain). The probability of this is $(0.4)(0.6)$.

For $X=2$, we need RRN. The probability of this is $(0.4)^2(0.6)$.

And so on.

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