Question

Approximately 80,000 marriages took place in the state of New York last year. Estimate the probabilitythat for at least one of these couples,(a) both partners were born on April 30;(b) both partners celebrated their birthday on the same day of the year.State your assumptions.

Answer 1

a. 0.45 b. 1

Explanation:

We will be using Poisson Approximation of Binomial because n = 80,000 is large and probability (p) is very small.

We calculate for (a) as follows:

The probability that both partners were born on April 30 is

p = 1/365 X 1/365

p = 1/133,225

p = 0.00000751

Using Poisson Approximation, we have:

λ = np

λ = 80,000 X 0.00000751

λ = 0.6

We use λ to calculate thus:

P (X
`\geq` 1) = 1 - P ( X = 0)

= 1 - e^-λ

= 1 - e^-0.6

= 0.451

There is a 45.1% probability that, for at least one of these couples, both partners were born on April 30.

(b) To calculate the probability that both partners celebrated their birthday on the same day:

p (same birthday) = 365 X 1/365 X 1/365

= 1/365

λ = np

λ = 80,000 X 1/365

λ = 219.17

P (X
`\geq` 1) = 1 - P ( X = 0)

= 1 - e^-λ

= 1 - e^-219.17

≈ 1

There is almost 100% probability that, for at least one of these couples, both partners celebrate their birthday on the same day of the year.