Answer:
a) player’s expected payoff is $ 240
b) probability the player loses $1000 or more is 0.1788
c) probability the player wins is 0.3557
d) probability of going broke is 0.0594
Step-by-step explanation:
Given:
Since there are 60 hands per hour and the player plays for four hours then the sample size is:
n = 60 * 4 = 240
The player’s strategy provides a probability of .49 of winning on any one hand so the probability of success is:
p = 0.49
a)
Solution:
Expected payoff is basically the expected mean
Since the bet is $50 so $50 is gained when the player wins a hand and $50 is lost when the player loses a hand. So
Expected loss = μ
= ∑ x P(x)
= 50 * P(win) - 50 * P(lose)
= 50 * P(win) + (-50) * (1 - P(win))
= 50 * 0.49 - 50 * (1 - 0.49)
= 24.5 - 50 ( 0.51 )
= 24.5 - 25.5
= -1
Since n=240 and expected loss is $1 per hand then the expected loss in four hours is:
240 * 1 = $ 240
b)
Using normal approximation of binomial distribution:
n = 240
p = 0.49
q = 1 - p = 1 - 0.49 = 0.51
np = 240 * 0.49 = 117.6
nq = 240 * 0.51 = 122.5
both np and nq are greater than 5 so the binomial distribution can be approximated by normal distribution
Compute z-score:
z = x - np / √(np(1-p))
= 110.5 - 117.6 / √117.6(1-0.49)
= −7.1/√117.6(0.51)
= −7.1/√59.976
= −7.1/7.744417
=−0.916789
Here the player loses 1000 or more when he loses at least 130 of 240 hands so the wins is 240-130 = 110
Using normal probability table:
P(X≤110) = P(X<110.5)
= P(Z<-0.916)
= 0.1788
c)
Using normal approximation of binomial distribution:
n = 240
p = 0.49
q = 1 - p = 1 - 0.49 = 0.51
np = 240 * 0.49 = 117.6
nq = 240 * 0.51 = 122.5
both np and nq are greater than 5 so the binomial distribution can be approximated by normal distribution
Compute z-score:
z = x - np / √(np(1-p))
= 120.5 - 117.6 / √117.6(1-0.49)
= 2.9/√117.6(0.51)
= 2.9/√59.976
= 2.9/7.744417
=0.374463
Here the player wins when he wins at least 120 of 240 hands
Using normal probability table:
P(X>120) = P(X>120.5)
= P(Z>0.3744)
= 1 - P(Z<0.3744)
= 1 - 0.6443
= 0.3557
d)
Player goes broke when he loses $1500
Using normal approximation of binomial distribution:
n = 240
p = 0.49
q = 1 - p = 1 - 0.49 = 0.51
np = 240 * 0.49 = 117.6
nq = 240 * 0.51 = 122.5
both np and nq are greater than 5 so the binomial distribution can be approximated by normal distribution
Compute z-score:
z = x - np / √(np(1-p))
= 105.5 - 117.6 / √117.6(1-0.49)
= -12.1/√117.6(0.51)
= -12.1/√59.976
= -12.1/7.744417
=−1.562416
Here the player loses 1500 or more when he loses at least 135 of 240 hands so the wins is 240-135 = 105
Using normal probability table:
P(X≤105) = P(X<105.5)
= P(Z<-1.562)
= 0.0594
is called an "exponential function",