Consider n people and suppose that each of them has a birthday that is equally likely...

Consider n people and suppose that each of them has a birthday that is equally likely to be any of the 365 days of the year. Furthermore, assume that their birthdays are independent, and let A be the event that no two of them share the

same birthday. Define a “trial” for each of the ?n? pairs of people and say that 2

trial (i, j ), I ̸= j, is a success if persons i and j have the same birthday. Let Si, j be the event that trial (i, j) is a success.

(a) Find P(Si,j), i ̸= j.
(b) Are Si, j, and Sk,r independent when i, j, k, r are all distinct?
(c) Are Si, j and Sk, j independent when i, j, k are all distinct?
(d) Are S1,2, S1,3, S2,3independent?
(e) Employ the Poisson paradigm to approximate P(A).
(f) Show that this approximation yields that P(A) ≈ .5 when n = 23.
(g) Let B be the event that no three people have the same birthday. Approximate the value of n that makes P(B) ≈ .5. (Whereas a simple combinatorial argument explicitly determines P(A), the exact determination of P(B) is very

complicated.)

Hint: Define a trial for each triplet of people.

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Expert Solution

orchestra answered 1 year ago

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