A roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers chooses red or black. A bet of $1 on red returns $2 if the ball lands in a red slot. Otherwise, the player loses his dollar. When gamblers bet on red or black, the two green slots belong to the house. Because the probability of winning $2 is 18/38, the mean payoff from a $1 bet is twice 18/38, or 94.7 cents. Explain what the law of large numbers tells us about what will happen if a gambler makes very many bets on red.

Law of large number states that if an experiment is repeated independently many times, the expected value should be close to the average. In law of large numbers, as the sample size grows, the mean will get closer to the total number.

In this case, we are told that the mean payoff from a $1 bet is 94.7 cents. Since, the mean payoff is 94.7 cents, the long run average on each lost bet will be 5.3 cents (100-94.7). From the law of large numbers, it can be expected that in the long run a gambler will lose a total of 5.3 cents multiplied by the total the number of bets placed. It means that, if a gambler places a bet on red 800 times, expecting that after 800 spins he loses all 800 times we i. e 800*5.3cents = $42.49