In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2... 36

Question

In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2... 36 To play too game a metal ball ts spun around toe wheel and w allowed to tan into one of the numbered slots It the number of the slot the ball tails mto matches the number you selected, you win 535. otherwise you lose 51 Complete parts (a) toroogh (g) below What is the probability of being ahead alter playing the game 90 times'? that is. what is the probability that the sample mean is greater than 0 tor n = 90? What is the probability of being ahead alter playing the game 270 times'' What is the probability of being ahead alter playing the game 900 times? The probability of being ahead decreases as the number of games played decreases The probability of being ahead decreases as the number of games played increases The probability of being ahead increases as the number of games played Increases The probability of being ahead remains the same as the number of games played vanes

Explanation / Answer

Note that for this game,
  
E(x) = Expected value = mean = Sum(xP(x)) =    -0.052631579

Var(x) = E(x^2) - E(x)^2 =    33.20775623

s(x) = sqrt [Var(x)] = 5.762617134

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d)

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 =    90      
s = standard deviation =    5.762617134      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.086645874      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.086645874   ) =    0.4654765 [ANSWER]

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e)

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 =    270      
s = standard deviation =    5.762617134      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.150075056      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.150075056   ) =    0.4403527 [ANSWER]

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f)

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 =    900      
s = standard deviation =    5.762617134      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.273998312      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.273998312   ) =    0.392042959 [ANSWER]

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g)

As we can see, it is

OPTION B: The probability of being ahead decreases as the number of games played increases. [ANSWER]

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