Question

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

Answer 1

a)0,45119

b)1

Explanation:

For part A of the problem we must first find the probability that both people in the couple have the same birthday (April 30)

`P=(1)/(365) *(1)/(365)=(1)/(133225) \\`

Now the poisson approximation is used

λ=nP=80000*1/133225=0,6

Now, let X be the number of couples that birth April 30

P(X ≥ 1) =

1 − P(X = 0) =

`1-((e^-0.6)*(-0,6)^(0) )/(0!)`

P(X ≥ 1) = 0,45119

B) Now want to find the

probability that both partners celebrated their birthday on th, assuming that the year is 52 weeks and therefore 52 thursday

`P=52*(1)/(365) *(1)/(365)=(52)/(133225) \\`

Now the poisson approximation is used

λ=nP=80000*52/133225=31.225

Now, let X be the number of couples that birth same day

P(X ≥ 1) =

1 − P(X = 0) =

`1-((e^-31.225)*(-31.225)^(0) )/(0!)`

P(X ≥ 1) = 1