Question

A roulette wheel has 38 slots: 18 red, 18 black, and 2 green. The ball ends at one of these at random.

You are a player who plays a large number of games and makes an even bet of Ghc 1 on red in every

game. After n games, what is the probability that you are ahead? Answer this for n = 100 and

n = 1000.

Answer 1

Let $S_n$ be the number of times you win. This is a Binomal $(n,\dfrac{9}{19})$ random variable.

P(ahead)=P(win more than half of the games)

     $=P(S_n>\dfrac{n}{2}) \\[9pt] =P(\dfrac{S_n-np}{\sqrt{np(1-p)}}>\dfrac{\frac{n}{2}-np}{\sqrt{np(1-p)}} ) \\[9pt] =P(Z>\dfrac{(\frac{1}{2}-p)\sqrt{n}}{\sqrt{p(1-p)}})$

  

For n=100,

$P(Z>\dfrac{5}{\sqrt{90}}=P(Z>0.527)=0.2990$

For n=1000.

$P(Z>\dfrac{5}{3})=P(Z>1.66)=0.0478$