Answer:
0.9999
Step-by-step explanation:
Let X be the random variable that measures the time that a switch will survive.
If X has an exponential distribution with an average life β=44, then the probability that a switch will survive less than n years is given by
[tex]\bf P(X<n) = 1-e^{-n/44}[/tex]
So, the probability that a switch fails in the first year is
[tex]\bf P(X<1) = 1-e^{-1/44}=0.02247[/tex]
Now we have 100 of these switches installed in different systems, and let Y be the random variable that measures the the probability that exactly k switches will fail in the first year.
Y can be modeled with a binomial distribution where the probability of “success” (failure of a switch) equals 0.0225 and
[tex]\bf P(Y=k)=\binom{100}{k}(0.02247)^k(1-0.02247)^{100-k}[/tex]
where
[tex]\bf \binom{100}{k}[/tex] equals combinations of 100 taken k at a time.
The probability that at most 15 fail during the first year is
[tex]\bf \sum_{k=0}^{15}\binom{100}{k}(0.02247)^k(1-0.02247)^{100-k}=0.9999[/tex]
