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

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pairs of people and say that 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) ≈ 0.5 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)≈0.5. (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.