Respuesta :
Answer:
There is a 75.65% probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week.
Step-by-step explanation:
For each teenager, there are only two possible outcomes. Either they watched a rented video at least once during a week, or they did not. This means that we can solve this problem using the binomial probability distribution.
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 combinatios 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:
The probability that a randomly selected teenager watched a rented video at least once during a week was 0.75. This means that [tex]p = 0.75[/tex].
What is the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week?
Group of 7, so [tex]n = 7[/tex].
[tex]P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7)[/tex].
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{7,5}.(0.75)^{5}.(0.25)^{2} = 0.3115[/tex]
[tex]P(X = 6) = C_{7,6}.(0.75)^{6}.(0.25)^{1} = 0.3115[/tex]
[tex]P(X = 7) = C_{7,7}.(0.75)^{7}.(0.25)^{0} = 0.1335[/tex]
So
[tex]P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) = 0.3115 + 0.3115 + 0.1335 = 0.7565[/tex].
There is a 75.65% probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week.
