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!
Best Answer
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.