Consider a group of n people.
(a) Explain why the pattern below gives the probabilities that the n people have distinct birthdays.
n = 2: 365 / 365 · 364 / 365 = 365 · 364 / 3652
n = 3: 365/ 365 · 364 / 365 · 363 / 365 = 365 · 364 · 363 / 3653
(b) Use the pattern in part (a) to write an expression for the probability that n = 4 people have distinct birthdays.
(c) Let Pn be the probability that the n people have distinct birthdays. Verify that this probability can be obtained recursively by

(d) Explain why Qn = 1 ˆ’ Pn gives the probability that at least two people in a group of n people have the same birthday.
(e) Use the results of parts (c) and (d) to complete the table.

(f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than 1 / 2? Explain.

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365 – (n – 1), 365 P, P = 1 and P, n-1 15 | 20 | 23 30 | 40 | 50 п 10 P,