Respuesta :
Answer:
a) The range of the random variable is 0,1,2,3.
b) There is a 38.4% probability that exactly only two of the three people will be alive at age 65.
Step-by-step explanation:
For each person aged 20, there are only two possible outcomes. Either they will be alive, or they will not be alive. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
80% chance that a person aged 20 will be alive at age 65, which means that [tex]p = 0.8[/tex]
Three people aged 20 are selected at random, which means that [tex]n = 3[/tex]
a). let x (random variable) denote the number of people of the three who are alive at age 65. what is the range of the random variable?
The range is the possible number of people in the sample who will be alive.
So the range of the random variable is 0,1,2,3.
b). find the probability that exactly only two of the three people will be alive at age 65.
This is P(X = 2).
So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 2) = C_{3,2}.(0.8)^{2}.(0.2)^{1} = 0.384[/tex]
There is a 38.4% probability that exactly only two of the three people will be alive at age 65.
