Question

1.

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 0.29. Find the probability that an average player wins

A. twice in 5 hands.

Probability =

B. 11 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 0.43. Assuming that the player plays the basic strategy, find the probability that he or she wins

C. twice in 5 hands.

Probability =

D. 11 or more times in 26 hands.

Probability =

2.

A game of chance involves rolling an unevenly balanced 4-sided die. The probability that a roll comes up 1 is 0.2, the probability that a roll comes up 1 or 2 is 0.51, and the probability that a roll comes up 2 or 3 is 0.46 . If you win the amount that appears on the die, what is your expected winnings? (Note that the die has 4 sides.)

3.

The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.9 per week. Find the probability of the following events.

A. No accidents occur in one week.

Probability =

B. 3 or more accidents occur in a week.

Probability =

C. One accident occurs today.

Probability =

Answer 1

Answer:

1.

Given,

p = 0.29

Binomial distribution P(X) = nCr*p^r*q^(n-r)

nCr = n!/(n-r)!*r!

a)

P(X = 2) = 5C2*0.29^2*(1-0.29)^3

= 0.3010

b)

11 or more time in 26 hands

P(11 <= X <= 26) = 1 - P(0 <= X <= 10)

= 1 - [P(0) + P(1) + P(2) + P(3) + .......... + P(10)]

= 1 - [26C0*0.29^0*0.71^26 + 26C1*0.29^1*0.71^25 + 26C2*0.29^2*0.71^24 + ........... + 26C10*0.29^10*0.71^16]

= 1 - [0.0001 + 0.0014 + 0.0074 + 0.0240 + 0.0565 + 0.1015 + 0.1451 + 0.1694 + 0.1643 + 0.1342 + 0.0932]

= 1 - 0.8971

= 0.1029

c)

p = 0.43

P(X = 2) = 5C2*0.43^2*(1-0.43)^3

= 0.3424

d)

11 or more time in 26 hands

P(11 <= X <= 26) = 1 - P(0 <= X <= 10)

= 1 - [P(0) + P(1) + P(2) + P(3) + .......... + P(10)]

= 1 - [26C0*0.43^0*0.57^26 + 26C1*0.43^1*0.57^25 + 26C2*0.43^2*0.57^24 + ........... + 26C10*0.43^10*0.57^16]

= 1 - [0 + 0 + 0.0001 + 0.0005 + 0.0022 + 0.0072 + 0.0191 + 0.0411 + 0.0737 + 0.1111 + 0.1425]

= 1 - 0.3975

= 0.6025

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