The probability that Cayuga Lake will freeze

Question

For my statistics homework, I need to calculate the probability that a lake will freeze next in exactly 5 years, and in 25 or more years. The probability that it will freeze in any given year is 0.0435 and I know that I need to use the binomial distribution to calculate this probability. My issue is that my probability that I come up with is only the probability that it will freeze once in 5 or 25 years, not that it will freeze next in exactly 5 years or that it will freeze next after 25 years have passed. I attempted to calculate it using the probability of A given B but wasn't sure how to calculate the individual parts, or the probability of A and B intersect and the probability that it will not freeze during the first four years.

Answer 1

For 5 years wouldn't it be something along the lines of the probability of not freezing in the next 4 years and then freezing in the fifth?

$$(1-p)^4 p$$

For the 25 or more year part you might look at something similar: $$ (1-p)^{24}(1-(1-p)^N)$$

Where $N$ is some large number of years you are willing to wait after year 25 giving you: $$ (1-p)^{24}$$

Answer 2

Let n (= 5 or 26) be the first year that it freezes. The probability is given by n-1 years of not freezing by one year of freezing. Thus $P_n=.9565^{n-1}\times .0435$.

If the freezing for the second question involves 25 years of not freezing followed by freezing eventually then $P=.9565^{25}$, since the probability it will freeze eventually after 25 years $=1$