Question

The​ life, in​ years, of a certain type of electrical switch has an exponential distribution with an average life β=44. If 100 of these switches are installed in different​ systems, what is the probability that at most 15 fail during the first​ year?

Answer 1

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

$\bf P(X$

So, the probability that a switch fails in the first year is

$\bf P(X$

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  

$\bf P(Y=k)=\binom{100}{k}(0.02247)^k(1-0.02247)^{100-k}$

where  

$\bf \binom{100}{k}$ equals combinations of 100 taken k at a time.

The probability that at most 15 fail during the first year is

$\bf \sum_{k=0}^{15}\binom{100}{k}(0.02247)^k(1-0.02247)^{100-k}=0.9999$