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Probability

Question
CBSEENMA12033832
Wired Faculty App

Two cards are drawn simultaneously (or successively, without replacement) from a well-shuffled deck of 52 cards. Compute σ2 for the number of aces.

Solution

Let X denote the number of aces obtained in two draws. X takes the values 0, 1, 2.
Let p denote probability of first ace.
therefore space space space straight p space equals space 4 over 52 comma space space straight q space equals space 48 over 52
straight P left parenthesis straight X space equals space 0 right parenthesis space equals space 48 over 52 cross times 47 over 51 space equals 12 over 13 cross times 47 over 51 space equals space 188 over 221 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space 4 over 52 cross times 48 over 51 plus 48 over 52 cross times 4 over 51 space equals 16 over 221 plus 16 over 221 space equals space 32 over 221 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space 4 over 52 cross times 3 over 51 space equals space 1 over 31 cross times 1 over 17 space equals 1 over 221
∴      probability distribution is

therefore space space space space straight X space takes space the space values space 0 comma space 1 comma space 2 space with space probabilities space 188 over 221 comma space 32 over 221 comma space 1 over 221

straight mu space equals sum from blank to blank of straight x space straight p subscript straight i space equals space 34 over 221 straight sigma squared space equals sum from blank to blank of straight x squared space straight p space subscript straight i space minus space straight mu squared space equals space 36 over 221 space minus space open parentheses 34 over 221 close parentheses squared space equals space 36 over 221 minus open parentheses 2 over 13 close parentheses squared space equals 36 over 221 minus 4 over 169 space space space space space equals space fraction numerator 6084 minus 884 over denominator 37349 end fraction space equals 5200 over 37349 space equals 400 over 2873

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