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Alvin's database of friends contains n entries, but due to
Chapter , Problem 30(choose chapter or problem)
Alvin's database of friends contains n entries, but due to a software glitch, the addresses correspond to the names in a totally random fashion. Alvin writes a holiday card to each of his friends and sends it to the (software-corrupted) address. What is the probability that at least one of his friends will get the correct card? Hint: Use the inclusion-exclusion formula. Solution. Let Ak be the event that the kth card is sent to the correct address. We have for any k, j, i, etc., and P(Ak) = .!. = (n - I)! , n n! 1 1 (n - 2) ' P(Ak n Aj) = P(Ak)P(AJ I Ak) = - . -- = , ' . n n - 1 n. 1 1 1 (n - 3)! P(Ak n A) n Ai) = - . --. -- = n n - I' 1 n -2 n. Applying the inclusion-exclusion formula, P (Uk=IAk) = L P(Ai) - L P(Ail n Ai2 ) iESI (i1 ,i2)es2 + we obtain the desired probability P(Uk=I Ak) = ( n ) (n - I)! _ ( n) (n - 2)! + ( n ) (n - 3)! _ ... + (-lr-l 1 n! 2 n! 3 n! n! = 1 - + -"'+(-lr-l 2! 3! n! ' When n is large, this probability can be approximated by 1 - e-l.
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QUESTION:
Alvin's database of friends contains n entries, but due to a software glitch, the addresses correspond to the names in a totally random fashion. Alvin writes a holiday card to each of his friends and sends it to the (software-corrupted) address. What is the probability that at least one of his friends will get the correct card? Hint: Use the inclusion-exclusion formula. Solution. Let Ak be the event that the kth card is sent to the correct address. We have for any k, j, i, etc., and P(Ak) = .!. = (n - I)! , n n! 1 1 (n - 2) ' P(Ak n Aj) = P(Ak)P(AJ I Ak) = - . -- = , ' . n n - 1 n. 1 1 1 (n - 3)! P(Ak n A) n Ai) = - . --. -- = n n - I' 1 n -2 n. Applying the inclusion-exclusion formula, P (Uk=IAk) = L P(Ai) - L P(Ail n Ai2 ) iESI (i1 ,i2)es2 + we obtain the desired probability P(Uk=I Ak) = ( n ) (n - I)! _ ( n) (n - 2)! + ( n ) (n - 3)! _ ... + (-lr-l 1 n! 2 n! 3 n! n! = 1 - + -"'+(-lr-l 2! 3! n! ' When n is large, this probability can be approximated by 1 - e-l.
ANSWER:Step 1 of 5
Given Information:
Alvin’s database of friends contains n entries, but because of the software glitch the address corresponds to the names is in random fashion. When Alvin sends the holiday card to every entry from the database where software is corrupted. The main goal is to find the probability of at least one of his friends getting the correct card.