13 13131313 54 13 13 13 13 the number of ways of

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Unformatted text preview: -card deck, dealt to 4 players. Find P (each player gets an “Ace"). Size of the sample space (possible combination of “hands") is 52 13 52 − 13 13 26 52! = 13 13!13!13!13! 54 . 13, 13, 13, 13 The number of ways of distributing the four “Ace"s: 4 · 3 · 2. The number of ways of distributing the remaining 48 cards: 48! 12!12!12!12! Therefore, the final answer is: P (each player gets an “Ace") = M. Chen (IE@CUHK) ENGG2430C lecture 4 48! 4 · 3 · 2 12!12!12!12! 52! 13!13!13!13! . 15 / 19 Independent Trials and the Binomial Probabilities * Independent trials: a sequence of independent but identical stages Bernoulli trials: only two possible results at each stage Example: Toss a fair coin n times: a sequence of n Bernoulli trials P (H ) = p P (a sequence) = p # of heads (1 − p )# of tails P (HTTHHH ) =? P (k heads) = ∑ P (a sequence that contains k heads) k -head sequences = (# of k -head sequences) · p k (1 − p )n−k nk = p (1 − p )n−k k M. Chen (IE@CUHK) ENGG2430C lecture 4 16 / 19 Communication over BSC * Crossover probability: P (0r |1t ) = P (1r |0t ) = ε = 0.2. Assume P (0t ) = P (1t ) = 0.5. We transmit a bit over the BSC once and receive an 1. What is probability that 1 is transmitted? (by Bayes’ law) P (1t |1r ) = P (1r |1t ) M. Chen (IE@CUHK) ENGG2430C lecture 4 P (1t ) = 1 − ε = 0.8. P (1r ) 17 / 19 Communication over BSC - Repetition Coding * Crossover probability: P (0r |1t ) = P (1r |0t ) = ε = 0.2. Assume P (0t ) = P (1t ) = 0.5. We transmit the same bit over the BSC five times, and receive four 1’s and one 0’s. What is probability that 1 is transmitted? Define B ={ we receive four 1’s and one 0’s out of five Bernoulli trials.} We have P (1t |B ) = 5 (1 − ε )4 ε × 4 M. Chen (IE@CUHK) 5 4 (1 − ε )4 ε 0.5 × 0.5 + ENGG2430C lecture 4 5 4 ε 4 (1 − ε ) × 0.5 ≈ 0.985. 18 / 19 Thank You Reading: Ch. 1.4 and 1.5 of the textbook. Next lecture: Review I and Discrete Random Variable. Happy Lunar New Year! M. Chen (IE@CUHK) ENGG2430C lecture 4 19 / 19...
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This document was uploaded on 03/31/2014.

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