Probability of rain Say the probability of rain is $r$, and I wanted to find the probability that it will rain in two consecutive days in a month (assume the month has 30 days). We will also assume that the probability of rain is independent of all other days.
My intuition would tell me that the probability of raining in two consecutive days is $29r^2$ since $r^2$ is the probability of two consecutive days, but there are $29$ possible 'two consecutive days'. 
My issue now is that if we choose $r$ to be 'large enough', then this probability CAN exceed $1$, which is of course nonsensical.
My question is, what is wrong with my logic here?
 A: Denote by $p_0(n)$   the probability that after $n$ days you have not yet seen two rainy days in succession, and that it has been dry on day $n$. Similarly, denote by $p_1(n)$  the probability that after $n$ days you have not yet seen two rainy days in succession, but that it has rained on day $n$. The probability you are after is then given by $P=1-p_0(30)-p_1(30)$.
We  have the initial conditions $p_0(0)=1$, $p_1(0)=0$. Furthermore there is the recursion
$$\left.\eqalign{p_0(n)&=(1-r)\bigl(p_0(n-1)+p_1(n-1)\bigr)\cr p_1(n)&=r\>p_0(n-1)\cr}\right\}\qquad(n\geq1\ .$$
We can eliminate $p_1$ from these formulas and then get
$$p_0(n)-(1-r)p_0(n-1)-r(1-r)p_0(n-2)=0\qquad(n\geq2)\ .$$
This difference equation can be explicitly solved using the "Master Theorem". Unfortunately the resulting characteristic values contain square roots, so that the end result will not look nice. 
A: You ignore the problem of double-counting. There could be rain multiple times on consecutive days (such as on days 1 and 2 and also on days 4 and 5). You count that possibility twice in your calculation, so your resulting value is too high. Taking the multiple-counting into account reduces the formula and keeps it below one.
A better way to solve this problem uses a Markov chain with three states. State 1 is "we have not yet had two consecutive days with rain this month and the day just finished is not counted as having rain." State 2 is "we have not yet had two consecutive days with rain this month and the day just finished is counted as having rain." State 3 is "we have had two consecutive days with rain this month" which is clearly an absorbing state. Find the transition matrix for each day, then use that to find the transition matrix after 30 days. The probability of ending in state 3 is your final answer.
If you don't understand Markov chains, this problem is too advanced for you now.
