Of all the registered automobiles in a city, 5% fail the emissions test. Fifteen automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places.Part 1 of 4(a) Find the probability that exactly four of them fail the test.
Question
Of all the registered automobiles in a city, 5% fail the emissions test. Fifteen automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places.Part 1 of 4(a) Find the probability that exactly four of them fail the test.
Solution
This 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 k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- n is the number of trials
- k is the number of successes
In this case:
- n = 15 (the number of automobiles selected)
- k = 4 (the number of automobiles that fail the test)
- p = 0.05 (the probability that an automobile fails the test)
So, we can plug these values into the formula:
P(X=4) = C(15, 4) * (0.05^4) * ((1-0.05)^(15-4))
First, calculate C(15, 4), which is the number of ways to choose 4 automobiles out of 15. This is calculated as:
C(15, 4) = 15! / [4!(15-4)!] = 1365
Next, calculate (0.05^4), which is the probability of 4 automobiles failing the test:
(0.05^4) = 0.00000625
Then, calculate ((1-0.05)^(15-4)), which is the probability of the remaining 11 automobiles passing the test:
((1-0.05)^(15-4)) = 0.4632912303
Finally, multiply these three values together to get the probability:
P(X=4) = 1365 * 0.00000625 * 0.4632912303 = 0.0039370045
So, the probability that exactly four of the 15 randomly selected automobiles fail the emissions test is approximately 0.0039 or 0.39%.
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