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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.
 Spring '14

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