A gambler is planning to make a sequence of bets on a roulette wheel. Note that a roulette wheel has 38 numbers, of which 18 are red, 18 are black, and 2 are green. Each time the wheel is spun, each of the 38 numbers is equally likely to occur. The gambler will choose one of the following two sequences. Single-number bet: The gambler will bet $5 on a particular number before each spin. He will win a net amount of $175 if that number comes up and lose $5 otherwise. Color bet: The gambler will be $5 on the red color before each spin. He will win a net amount of $5 if a red number comes up and lose $5 otherwise.
a. If the gambler makes a sequence of 25 bets, which of the two betting schemes offers him a better chance of coming out ahead (winning more money than losing) after the 25 bets?
b. Now compute the probability of coming out ahead after 25 single-number bets of $5 each and after 25 color bets of $5 each. Do these results confirm your guess in part a? (Before using an approximation to find either probability, be sure to check whether it is appropriate.)