Question

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 seven. 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. (a) P(exactly five) (Round to four decimal places as needed.)

Answer 1

Let X be the number of births per minute in a country in a recent year

X ~ Poisson ( 7)

a) The probability that exactly five births in any given minute, i.e. P(X=5)

b)

$P(X \geq 5) =1- P(X<5)$

$P(X=0) =\frac{e^{-7}7^{0}}{0!} = 0.00091$

$P(X=1) =\frac{e^{-7}7^{1}}{1!} = 0.00638$

$P(X=2) =\frac{e^{-7}7^{2}}{2!} = 0.02234$

$P(X=3) =\frac{e^{-7}7^{3}}{3!} = 0.05213$

$P(X=4) =\frac{e^{-7}7^{4}}{4!} = 0.09123$

$P(X \geq 5) =1- 0.17299 =0.8270$

c)

$P(X\leq 5) =0.00091+0.00638+0.02234+0.05213+0.09123+0.12772=0.30071$

$P(X>5) = 1-P(X\leq 5) =1- 0.30071= 0.69929\approx 0.6993$