Among companies doing highway or bridge construction, 80% test employees for substance abuse. A study involves the random selection of 10 such companies. Find the probability that exactly 5 of the 10 companies test for substance abuse.
To find the probability that exactly 5 of the 10 companies test for substance abuse, we can use the binomial probability formula.
The formula for binomial probability is:
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes in n trials.
C(n,k) is the number of combinations of n items taken k at a time.
p is the probability of success in a single trial.
(1-p) is the probability of failure in a single trial.
n is the total number of trials.
In this case:
k = 5 (we want exactly 5 companies to test for substance abuse)
n = 10 (there are 10 companies)
p = 0.80 (the probability that a company tests for substance abuse)
Substituting the values into the formula, we have:
P(X = 5) = C(10,5) * 0.80^5 * (1 - 0.80)^(10 - 5)
Calculating the combinations, we have:
C(10,5) = 10! / (5! * (10-5)!) = 252
Substituting this value back into the formula, we have:
P(X = 5) = 252 * 0.80^5 * (1 - 0.80)^(10 - 5)
Calculating this expression, we find:
P(X = 5) ≈ 0.0264
Therefore, the probability that exactly 5 of the 10 companies test for substance abuse is approximately 0.0264, or 2.64%.
To find the probability that exactly 5 out of the 10 randomly selected companies test for substance abuse, we can use the binomial probability formula.
The binomial formula is given by:
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 selecting k items from a set of n items
p is the probability of success in a single trial
q is the probability of failure in a single trial (1-p)
n is the total number of trials or observations
In this case:
n = 10 (the number of companies selected)
k = 5 (the number of companies that test for substance abuse)
p = 0.8 (the probability that a company tests for substance abuse)
Using the formula:
P(X=5) = C(10, 5) * (0.8)^5 * (1-0.8)^(10-5)
C(10,5) = 10! / (5!(10-5)!) = 252 (combinations)
P(X=5) = 252 * (0.8)^5 * (0.2)^5
Calculating this expression gives the probability that exactly 5 out of 10 companies test for substance abuse.