Question

A survey reported that 5% of Americans are afraid of being alone in a house at night. If a random sample of 20 Americans is selected, what is the probability that exactly 3 people in the sample are afraid of being alone at night.

Answer 1

To find the probability that exactly 3 people out of a sample of 20 Americans are afraid of being alone at night, we can use the binomial probability formula. This formula takes into account the probability of success on each trial, the number of trials, and the number of successful trials. Plugging in the values, we can calculate the desired probability.

Step-by-step explanation:

In this case, we are dealing with a binomial distribution, where each American surveyed can be classified as either afraid of being alone at night or not afraid. The probability of an American being afraid is 5%, which means the probability of an American not being afraid is 95%. Since we want to find the probability of exactly 3 people out of 20 being afraid, we can use the binomial probability formula:

P(X = k) = C(n,k) * p^k * (1-p)^(n-k)

where:

• 
  
• P(X = k) is the probability of exactly k successes
• 
  
• C(n,k) is the number of combinations of n objects taken k at a time
• 
  
• p is the probability of success on each trial
• 
  
• n is the number of trials
• 
  
• k is the number of successful trials

In this case, we have:

• 
  
• p = 0.05
• 
  
• (1-p) = 0.95
• 
  
• n = 20
• 
  
• k = 3

Plugging these values into the formula, we get:
P(X = 3) = C(20, 3) * (0.05)^3 * (0.95)^(20-3)
Calculating this expression gives us the probability that exactly 3 people in the sample of 20 are afraid of being alone at night.

Answer 2

5.96% probability that exactly 3 people in the sample are afraid of being alone at night.

Step-by-step explanation:

For each person, there are only two possible outcomes. Either they are afraid of being alone at night, or they are not. The probability of a person being afraid of being alone at night is independent of any other person. 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.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

5% of Americans are afraid of being alone in a house at night.

This means that
`p = 0.05`

If a random sample of 20 Americans is selected, what is the probability that exactly 3 people in the sample are afraid of being alone at night.

This is P(X = 3) when n = 20. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 3) = C_(20,3).(0.05)^(3).(0.95)^(17) = 0.0596`

5.96% probability that exactly 3 people in the sample are afraid of being alone at night.