The payout on matching a color (red/black) in roulette is one to one. In other words, if you bet a dollar and win, you get a dollar. There are 38 slots on a roulette wheel,18 of which are red, 18 of which are black, and 2 of which are green (0 and 00)
1. Compute the expected value for a $1 bet on the color red
2. Interpret your answer for part a in the context of the problem
1) To calculate the expected value, we need to determine the odds of winning, and the amount won if we win, and the odds of losing, and the amount lost if we lose. In the roulette question, the odds of winning are 18/38, that is, there are 18 red slots on the roulette wheel out of 38. The amount won if we land on a red is $1. We multiply the 18/38 odds of winning by the $1 gained, and get $0.4737. The odds of losing are 20/38, that is, there are 20 slots that are not red. The amount lost if we do not land on a red is our $1 bet. We multiply the 20/38 odds of losing by the -$1 (dollar lost), and get -$0.5263. Now we add the $0.4737 and the -$0.5263, and get -$0.0526. Thus, the expected value of a $1 bet is -$0.0526, or, rounded to the nearest penny, -$0.05.
2) The expected value is the long-term value we would expect to gain (or lose in this case) over multiple bets. The more bets that are made, the closer the actual winnings and losses would be to the expected value. That is, if we made one thousand separate $1 bets, we would expect to have $950 in our pockets and to have lost $50. We can expect to lose $0.05 cents per bet over the long run.