Question

suppose that the prevalence of a certain type of tree allergy is 0.42 in the general population. if 100 people randomly chosen from this population are tested for this allergy, what is the probability that exactly 42 of them will have this allergy? please write your answer as a decimal, precise to at least four decimal places.

Answer 1

The probability that exactly 42 out of 100 people randomly chosen from a population will have a certain type of tree allergy, given that the prevalence of the allergy is 0.42, is approximately 0.0806.

How to find probability?

To calculate this probability, use the binomial probability formula. The binomial probability formula is given by:

`\[ P(X = k) = \binom{n}{k} p^k (1-p)^(n-k) \]`

Where:

`\( P(X = k) \)` = probability of exactly k successes in n trials.

`\( \binom{n}{k} \)` = binomial coefficient, which can be calculated as
`\( (n!)/(k!(n-k)!) \)`.

n = number of trials (in this case, 100 people).

k = number of successes (in this case, 42 people with the allergy).

p = probability of success on a single trial (in this case, the prevalence of 0.42).

Given:

n = 100

k = 42

p = 0.42

The binomial coefficient represents the number of ways to choose k successes out of n trials. It is calculated as:

`\[ \binom{n}{k} = (n!)/(k! * (n - k)!) \]`

For n = 100 and k = 42:

`\[ \binom{100}{42} = (100!)/(42! * (100 - 42)!) \]`

The binomial probability formula is:

`\[ P(X = 42) = \binom{100}{42} * 0.42^(42) * (1 - 0.42)^(100 - 42) \]`

Using the binomial probability mass function (pmf) is as follows:

`\[ P(X = 42) = \binom{100}{42} * 0.42^(42) * (1 - 0.42)^(58) \]`

= 0.0806

The result is approximately 0.0806, precise to at least four decimal places.