Question

Based on a​ poll, 60​% of adults believe in reincarnation. Assume that 5 adults are randomly​ selected, and find the indicated probability. What is the probability that exactly 4 of the selected adults believe in​ reincarnation?

Answer 1

Using the binomial probability formula, we find the probability that exactly 4 out of 5 randomly selected adults believe in reincarnation to be 25.92%.

Step-by-step explanation:

The student's question pertains to the topic of probability, specifically a binomial probability where the success criterion is the belief in reincarnation by an adult, and the success probability (p) is 0.60. We are seeking the probability that exactly 4 out of 5 selected adults believe in reincarnation, which is a binomial probability problem.

The binomial probability formula is:

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

Where:

• P(X=k) is the probability of having exactly k successes in n trials
• C(n, k) is the combination of n things taken k at a time
• p is the success probability
• (1-p) is the failure probability
• n is the number of trials
• k is the number of successes we are looking for (which is 4 in this case)

To solve the problem:

C(5, 4) = 5!/(4!(5-4)!)= 5

Then calculate p^k, which is (0.60)^4

And also (1-p)^(5-4), which is (0.40)^1

Multiplying these together:

P(X=4) = 5 × (0.60)^4 × (0.40)^1

P(X=4) = 5 × 0.1296 × 0.40

P(X=4) = 0.2592 or 25.92%

Answer 2

There is a 25.92% probability that exactly 4 of the selected adults believe in​ reincarnation.

Step-by-step explanation:

For each adult, there are only two possible outcomes. Either they believe in reincarnation, or they do not believe. 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.

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

In which
`C_(n,x)` is the number of different combinatios 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.

In this problem

There are 5 adults, so
`n = 5`

60% believe in reincarnation, so
`p = 0.6`

What is the probability that exactly 4 of the selected adults believe in​ reincarnation?

This is P(X = 4).

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

`P(X = 4) = C_(5,4).(0.6)^(4).(0.4)^(1) = 0.2592`

There is a 25.92% probability that exactly 4 of the selected adults believe in​ reincarnation.