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19 October, 12:17

For a group of 50 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 3 people:

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  1. 19 October, 13:56
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    0.0177 days

    Step-by-step explanation:

    Solution:-

    - The group has n = 50 people

    - Each person is equally likely to have a birthday on each of 365 days in the year.

    - The probability of success p = 1 / 365

    - The probability of failure q = (1 - 1/365)

    - We will denote random variable X as a given day has k number of birthdays.

    - X follows binomial distribution:

    X ~ Bi (50, 1/365)

    - The probability that there would be 3 birthdays on a day would be:

    P (X = 3) = 50C3 (1 / 365) ^3 * (1 - 1/365) ^47

    P (X = 3) = 0.00035

    - The expected number of days for exactly three people to have same birth date:

    E (X = 3) = P (X=3) * n = 0.00035*50 = 0.0177 days
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