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 four ​(b) at least four and ​ (c) more than four.

Answer 1

a)

Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X = 4)
P(X = 4) = 7^4 * e^-7/4!
P(X = 4) = 0.0912
Ans: 0.0912

not unusual

b)
Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X > =4) = 1 - P(X <= 3).
P(X > =4 ) = 1 - (7^0 * e^-7/0!) + (7^1 * e^-7/1!) + (7^2 * e^-7/2!) + (7^3 * e^-7/3!)
P(X >=4) = 1 - (0.0009 + 0.0064 + 0.0223 + 0.0521)
P(X > =4) = 1 - 0.0817 = 0.9183

not unusual

c)

Here, λ = 7 and x = 4
As per Poisson's distribution formula P(X = x) = λ^x * e^(-λ)/x!

We need to calculate P(X > 4) = 1 - P(X <= 4).
P(X > 4) = 1 - (7^0 * e^-7/0!) + (7^1 * e^-7/1!) + (7^2 * e^-7/2!) + (7^3 * e^-7/3!) + (7^4 * e^-7/4!)
P(X > 4) = 1 - (0.0009 + 0.0064 + 0.0223 + 0.0521 + 0.0912)
P(X > 4) = 1 - 0.1729 = 0.8271

not unusual