Of all the registered automobiles in a city, 7% fail the emissions test. Thirteen automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places.Part: 0 / 40 of 4 Parts CompletePart 1 of 4(a) Find the probability that exactly four of them fail the test.
Question
Of all the registered automobiles in a city, 7% fail the emissions test. Thirteen automobiles are selected at random to undergo an emissions test. Round the answers to at least four decimal places.Part: 0 / 40 of 4 Parts CompletePart 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 on a single trial
- n is the number of trials
- k is the number of successes
In this case:
- n = 13 (the number of automobiles selected)
- k = 4 (the number of automobiles that fail the test)
- p = 0.07 (the probability that an automobile fails the test)
First, calculate C(n, k) = C(13, 4). This is the number of ways to choose 4 automobiles out of 13, which is 715.
Next, calculate p^k = 0.07^4 = 0.0002401.
Then, calculate (1-p)^(n-k) = (1-0.07)^(13-4) = 0.9339.
Finally, multiply these three values together to get the probability:
P(X=4) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 715 * 0.0002401 * 0.9339 = 0.1601.
So, the probability that exactly four of the 13 randomly selected automobiles fail the test is approximately 0.1601 or 16.01%.
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