# Conditional Probability with a Recursion Formula

The chance of a sunny day is $q$ independent of other days. Let $P_n$ be the probability that in $n$ days there are never two sunny days in a row.

Find $P_2$ and $P_3$ So the probability of in 2 days there are never two sunny days in a row is $q^cq^c$. But this only happens once out of four times, so we have a $1/4$ as this happens. My question here is, which one is correct. The probability of 3 days there are never 2 sunny days in a row is $q^cq^cq$ or $qq^cq^c$ or $q^cq^cq^c$, twice. This is out of a total of 12 ways. (6 ways to move around $q^cq^cq$ and 6 ways to move around $q^cqq$). So we have $4/12=1/3$. Again which probability am I looking for?

By conditioning on the outcome of the first day's weather, find a recursion formula for $P_n$ in terms of $P_{n-1}$ and $P_{n-2}$. I am completely lost on this one and have no idea where to begin, which I think is because I am unsure about the above part. But if someone could point me in the right direction that would be great.

Suppose q=.3. Use the formula to solve for prob of at least two sunny days in a row over a course of 7 days. For this question, is it as simple as plugging in the values? My argument against that, is the formula shows prob of 7 days there are never two sunny days. (so plug in to get this value). But one also needs to find the probability of one non sunny day (which is simply $q^cq^6*7$ over the sample space, which I believe is 5040, the permutation of 7 with 7. Then do 1 minus those two events to figure out this probability. Does this sound accurate?

For $P_2$: the chance of having two sunny days in a row is $q^2$. Therefore $$P_2=1-q^2\ .$$ For $P_3$: there are not two sunny days in a row when either of the following mutually exclusive events occurs:

• day 1 is not sunny and the following two days are not both sunny;
• day 1 is sunny, day 2 is not.

So $$P_3=(1-q)P_2+q(1-q)=(1-q)(1+q-q^2)=1-2q^2+q^3\ .$$

For $P_n$, there are not two sunny days in a row when any of the following mutually exclusive events occurs:

• day 1 is not sunny and there are never two consecutive sunny days in the following $n-1$ days;
• days 1 is sunny, day 2 is not, and there are never two consecutive in the following $n-2$.

Hence $$P_n=(1-q)P_{n-1}+q(1-q)P_{n-2}\ .$$

For the final part of the question, this is $$P_n=0.7P_{n-1}+0.21P_{n-2}\ .$$ There are methods for solving this kind of homogeneous linear second-order recurrence; if you have not learned these in class then I would imagine you are just supposed to do the arithmetic and calculate successively $P_2,P_3,\ldots,P_7$. Your final answer is then $1-P_7$.

• could you perhaps be a little clearer when talking about $P_3$? because as I read it as the prob that in 3 days there are never 2 sunny days in a row. The complement happens in two ways, first day is sunny and the rest are not or the last day is sunny and the previous ones were not. – Jack Armstrong Mar 4 '15 at 15:24
• Not sure I understand what you are saying in your comment. If $P_3$ is the probability that there are never two sunny days in a row, the complement of this is the event that there are (at least) two sunny days in a row. This can happen in three ways: SSS, SSN, NSS. (But in my calculation I did not use the complement, I found the probability directly.) – David Mar 4 '15 at 21:12