L14_Bday_print

L14_Bday_print - P ( A n ) = 1-P ( B n ) = 1-365 n 365 n 4...

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1 The Birthday Paradox Suppose 25 people are in a room. What is the probability that at least two of them share a birthday? Less than 1/2? Actually it’s greater than 1/2. We will see the analysis of the problem of calculating the probability of event: A n : There are n people in a room and at least two of them share a birthday. (We will assume that a year has 365 days and there are no twins in the room.) This will be very similar to the analysis of hashing n keys into a table of size 365.
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2 The Birthday Paradox Sample space S : All n -tuples ( l 1 , l 2 , ..., l n ) where the l i are birthdays, i.e. numbers in [1 , 365] . | S | = 365 n Now let B n be the event that no two people share a birthday. The number of such outcomes is 365 × 364 × ... × (365 - ( n - 1)) = 365 n So, by Theorem 5.12 seen in class, Note: Easy to calculate using P ( B 1 ) = 1 , P ( B n ) = 365 n 365 n P ( B n +1 ) = P ( B n ) × 365 - n 365
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3 The Birthday Paradox A n : At least two of the n people share a birthday. B n : No two of the n people share a birthday. A n and B n are complementary so
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Unformatted text preview: P ( A n ) = 1-P ( B n ) = 1-365 n 365 n 4 The Birthday Paradox n A n B n n A n B n 1 0.00000000 1.00000000 16 0.28360400 0.71639599 2 0.00273972 0.99726027 17 0.31500766 0.68499233 3 0.00820416 0.99179583 18 0.34691141 0.65308858 4 0.01635591 0.98364408 19 0.37911852 0.62088147 5 0.02713557 0.97286442 20 0.41143838 0.58856161 6 0.04046248 0.95953751 21 0.44368833 0.55631166 7 0.05623570 0.94376429 22 0.47569530 0.52430469 8 0.07433529 0.92566470 23 0.50729723 0.49270276 9 0.09462383 0.90537616 24 0.53834425 0.46165574 10 0.11694817 0.88305182 25 0.56869970 0.43130029 11 0.14114137 0.85885862 26 0.59824082 0.40175917 12 0.16702478 0.83297521 27 0.62685928 0.37314071 13 0.19441027 0.80558972 28 0.65446147 0.34553852 14 0.22310251 0.77689748 29 0.68096853 0.31903146 15 0.25290131 0.74709868 30 0.70631624 0.29368375...
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L14_Bday_print - P ( A n ) = 1-P ( B n ) = 1-365 n 365 n 4...

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