Question

the probability that a randomly selected 40 - year - old male will live to be 41 years old is 0.996739, according to the national vital statistics report, vol. 71, no. 1.
(a) what is the probability that two randomly selected 40 - year - old males will live to be 41 years old?
(b) what is the probability that eight randomly selected 40 - year - old males will live to be 41 years old?
(c) what is the probability that at least one of eight randomly selected 40 - year - old males will not live to be 41 years old? would it be unusual if at least one of eight randomly selected 40 - year - old males did not live to be 41 years old?

Explanation:

Step1: Identify probability of individual survival

The probability that a 40 - year - old male will live to be 41 years old is $p = 0.996739$.

Step2: Calculate probability for part (a)

For two independent 40 - year - old males, the probability that both will live to be 41 is $p\times p=0.996739\times0.996739 = 0.993491$.

Step3: Calculate probability for part (b)

The probability that a 40 - year - old male will not live to be 41 is $q=1 - 0.996739 = 0.003261$. The probability that all eight 40 - year - old males will live to be 41 is $p^{8}=0.996739^{8}\approx0.9747$.

Step4: Calculate probability for part (c)

The probability that all eight 40 - year - old males will not live to be 41 is $q^{8}=0.003261^{8}\approx0$. The probability that at least one will not live to be 41 is $1 - p^{8}=1 - 0.9747 = 0.0253$. Since $0.0253<0.05$, it would be unusual.

Answer:

(a) $0.993491$
(b) $0.9747$
(c) $0.0253$, and it would be unusual.