Probability a Random Variable will assume a value In a fictional town of ABC, the weather patterns on different days are all independent of each other. Assume that each given day in ABC is sunny with probability 30% and rainy with probability 70%. Let T denote the random variable representing the number of days you need to wait to see two sunny days in a row in ABC. For example, if it is sunny both tomorrow and the day after, the value of T will be 2. If it is sunny tomorrow, rains the day after, and is then sunny the two days after that, the value of T will be 4. What is the probability that T=4? So if the question was asking the probability of T=3, the first day would have to rain and the last two would be sunny, so the probability would be given by 0.7*0.3*0.3. For T = 4, the last two days are sunny but the first two can either be both rainy or have one rainy and one sunny. I'm not sure how to assign probability to that. Any help would be appreciated!

We make the (unreasonable) assumption of independence. The patterns that give $T=4$ are as follows: RRSS, SRSS. The probabilities of these patterns can be computed from the information we were given.