Question

For a group of 70 people, assuming that each person is equally likely to have a birthday on each of 365 days in the year, compute (a) The expected number of days of the year that are birthdays of exactly 4 people:

Answer 1

Answer:

0.0157

Step-by-step explanation:

From the information given:

The sample size  = 70

The expected no. of days of year that are birthday of exactly 4 people is:$P = \bigg [ \dfrac{1}{365} \bigg]^4$

The expected number of days with 4 birthdays = $\sum \limits ^{365}_{i=1} E(x_i)$

$\sum \limits ^{365}_{i=1} E(x_i) = 365 \times \bigg[ \ ^{70}C_{4} \times ( \dfrac{1}{365})^4 ( 1 - \dfrac{1}{365})^{70-4} \bigg]$

$\sum \limits ^{365}_{i=1} E(x_i) = 365 \times \bigg[ \ \dfrac{70!}{4!(70-4)!} \times ( \dfrac{1}{365})^4 ( 1 - \dfrac{1}{365})^{66} \bigg]$

$\sum \limits ^{365}_{i=1} E(x_i) = 365 \times \bigg[ \ 916895 \times 5.6342 \times 10^{-11} \times 0.8343768898 \bigg]$

= 0.0157

Therefore, the required probability = 0.0157