1 point) In the game of blackjack as played in casinos in Las Vegas, Atlantic Ci
ID: 3365933 • Letter: 1
Question
1 point) In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara Falls, as well as many other cities, the dealer has the advantage. Most players do not play very well. As a result, the probability that the average player wins a hand is about 0.46. Find the probability that an average player wins A. twice in 5 hands. Probability B. 9 or more times in 26 hands. Probability There are several books that teach blackjack players the "basic strategy" which increases the probability of winning any hand to 0.51. Assuming that the player plays the basic strategy, find the probability that he or she wins C. twice in 5 hands. Probability = D. 9 or more times in 26 hands. Probability =Explanation / Answer
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 5 * 0.46
= 2.3
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 5 * 0.46 * 0.54
= 1.242
III.
standard deviation = sqrt( variance ) = sqrt(1.242)
=1.114451
A.
P( X = 2 ) = ( 5 2 ) * ( 0.46^2) * ( 1 - 0.46 )^3
= 0.333194
B.
P( X < 9) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 26 8 ) * 0.46^8 * ( 1- 0.46 ) ^18 + ( 26 7 ) * 0.46^7 * ( 1- 0.46 ) ^19 + ( 26 6 ) * 0.46^6 * ( 1- 0.46 ) ^20 + ( 26 5 ) * 0.46^5 * ( 1- 0.46 ) ^21 + ( 26 4 ) * 0.46^4 * ( 1- 0.46 ) ^22 + ( 26 3 ) * 0.46^3 * ( 1- 0.46 ) ^23 + ( 26 2 ) * 0.46^2 * ( 1- 0.46 ) ^24 + ( 26 1 ) * 0.46^1 * ( 1- 0.46 ) ^25 + ( 26 0 ) * 0.46^0 * ( 1- 0.46 ) ^26
= 0.085362
P( X > = 9 ) = 1 - P( X < 9) = 0.9146
When X ~ B(n,0.51)
C.
P( X = 2 ) = ( 5 2 ) * ( 0.51^2) * ( 1 - 0.51 )^3
= 0.306005
D.
P( X < 9) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 26 8 ) * 0.51^8 * ( 1- 0.51 ) ^18 + ( 26 7 ) * 0.51^7 * ( 1- 0.51 ) ^19 + ( 26 6 ) * 0.51^6 * ( 1- 0.51 ) ^20 + ( 26 5 ) * 0.51^5 * ( 1- 0.51 ) ^21 + ( 26 4 ) * 0.51^4 * ( 1- 0.51 ) ^22 + ( 26 3 ) * 0.51^3 * ( 1- 0.51 ) ^23 + ( 26 2 ) * 0.51^2 * ( 1- 0.51 ) ^24 + ( 26 1 ) * 0.51^1 * ( 1- 0.51 ) ^25 + ( 26 0 ) * 0.51^0 * ( 1- 0.51 ) ^26
= 0.030102
P( X > = 9 ) = 1 - P( X < 9) = 0.9699
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