Homework SourseFind the indicated probabilities using the geometric distrib
Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities. The mean number of births per minute in a country in a recent year was about six. Find the probability that the number of births in any given minute is (a) exactly five, (b) at least five, and (c) more than five.
Solution
This is a POISSON DISTRIBUTION, as the mean number of oaccurrences per unit time is constant.
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a)
Note that the probability of x successes out of n trials is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 6
x = the number of successes = 5
Thus, the probability is
P ( 5 ) = 0.160623141 [ANSWER]
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b)
Note that P(at least x) = 1 - P(at most x - 1).
Using a cumulative poisson distribution table or technology, matching
u = the mean number of successes = 6
x = our critical value of successes = 5
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 4 ) = 0.2850565
Thus, the probability of at least 5 successes is
P(at least 5 ) = 0.7149435 [ANSWER]
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c)
Note that P(more than x) = 1 - P(at most x).
Using a cumulative poisson distribution table or technology, matching
u = the mean number of successes = 6
x = our critical value of successes = 5
Then the cumulative probability of P(at most x) from a table/technology is
P(at most 5 ) = 0.445679641
Thus, the probability of at least 6 successes is
P(more than 5 ) = 0.554320359 [ANSWER]
