Suppose that over the course of several days a gambler makes

Suppose that, over the course of several days, a gambler makes 3200 row bets in roulette, betting $1 each time. For a row bet, the gambler selects a row of 3 spaces out of the 38 possible spaces on a roulette table. If the gambler wins on one play, the gambler get his/her dollar back plus eleven more, for a net gain of $11. However, if the gambler loses, he/she loses $1. Find the probability that, after making these 3200 row bets, that the gambler wins at least $0. (Round your answers to four decimal places.)

Solution

Consider the table:

Thus,      
      
E(x) = Expected value = mean =    -0.052631579  
Var(x) = E(x^2) - E(x)^2 =    10.47091413  
s(x) = sqrt [Var(x)] =    3.235879189  

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0      
u = mean =    -0.052631579      
n = sample size =    3200      
s = standard deviation =    3.235879189      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.920087413      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.920087413   ) =    0.178763541 [ANSWER]

x	P(x)	x P(x)	x^2 P(x)
-1	0.921053	-0.92105	0.921053
11	0.078947	0.868421	9.552632