Q.4.53 - A First Course in Probability

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 the same day of the year. State your assumptions 

Step-by-Step Solution

Answer

(a) The probability that for at least one of these couples both partners were born on April 30 is 0.4512.

(b) The probability that for at least one of these couples both partners were born on same is 1 . 

Step 1: Given information (Part a)

Approximately 80,000 marriages took place in the state of New York last year. 

Step 2: Solution (Part a)

The probability that a person is born on April 30 is.

$P\left(\text{ a person is born on april }30\right)=\frac{1}{365}$

$P\left(\text{ Both persons born on April }30\right)=\frac{1}{365}\times \frac{1}{365}$

$=\frac{1}{(365{)}^2}$

This is tiny probability, so use Poisson approximation to binomial.

Mean;

$\lambda =\frac{80,000}{(365{)}^2}$

$=0.6$

Step 3: Final calculation (Part a)

Let X represents the numeral of couples that share April 30 as their birthday.

The probability that for at least one of these couples both partners were born on April 30 is, The probability function of the Poisson distribution is defined as,

$P(X=i)=e^{-\lambda }\frac{{\lambda }^i}{i!}i=0,1,2\dots .$

$\lambda =$parameter$P\left(\text{ no couple born on April }30\right)=P(X=0)$

$=\frac{e^{-0.6}(0.6{)}^0}{0!}$

$=e^{-0.6}$

$P\left(\text{ atleast one couple; both partners born on April }30\right)=1-e^{-0.6}$

$=0.4512$

Step 4: Final answer (Part a)

Thus, the probability that for at least one of these couples both partners were born on April 30 is 0.4512.

Step 5: Given information (Part b)

The probability that both partners celebrated their birthday on the same day of the year is, Now, find the probability of both partners celebrate their birthday on the same day, both the partners choose one of the 365 days of the year.

$P\left(\text{ Same birthday }\right)=\frac{365}{(365{)}^2}$

$=\frac{1}{365}$

Therefore the mean will be,

$\lambda =\frac{80,000}{365}$

$=219.178$

Step 6: Final calculation (Part b)

Let X denotes the number of couples that share same birthday.

The probability function of the Poisson distribution is defined as,

$P(X=i)=e^{-\lambda }\frac{{\lambda }^i}{i!}i=0,1,2\dots .$

$\lambda =\text{ parameter. }$

$P\left(\text{ no couple born on April }30\right)=P(X=0)$

$=\frac{e^{-219.178}(219.178{)}^0}{0!}$

$=e^{-219.178}$

$P\left(\text{ atleast one couple; both partners born on same day }\right)=1-e^{-219.178}$

$\approx 1$

Step 7: Final answer (Part b)

Therefore, the probability that for at least one of these couples both partners were born on same is 1 .