"The average temperature in the summer in a certain region is normally distributed with mean 72 and variance 4. Compute the probability that it will take at least 5 years before 3 summers have an average temperature exceeding 76."
My solution: We are given that μ=72 and σ^2=4 (or σ=2). I believe that we must integrate, from 5 to infinity, the function (1/Sqrt(2pi)(2))e^(-(x-72)^2/2(4)). I'm just unsure of how to interpret "before 3 summers have an average temperature exceeding 76".
$X \sim \mathcal{N}(72, 4)$.
Let $p = P(X > 76) = 1 - P(X < 76)$. (You can calculate this)
Define "success" to be the event $A = \{X > 76 \}$, so probability of success is $p$.
Then, the probability that it takes at least 5 "trials" to achieve 3 "successes" is:
$= 1 - \left [\text{probability of at least 3 successes in the first 4 trials } \right ]$
$ = 1 - \left [ p^3 + {3 \choose 1} p^3(1-p) \right ]$