Law of Large Numbers Statistics Example 2
Follow the full solution, then compare it with the other examples linked below.
Example 2
mediumA casino offers a game where you win \1 with probability 0.48 and lose \1 with probability 0.52. (a) What is the expected value per game? (b) Explain why the casino is guaranteed to profit in the long run using the law of large numbers.
Solution
- 1 Step 1: (a) E(X) = (1)(0.48) + (-1)(0.52) = 0.48 - 0.52 = -\0.04$ per game.
- 2 Step 2: (b) The expected value is negative for the player (β\0.04 per game). By the law of large numbers, the player's average loss per game converges to \0.04 as they play more games.
- 3 Step 3: Over millions of games, the casino earns approximately \$0.04 per game played. While individual players may win in the short run, the law of large numbers ensures the casino's average profit approaches the expected value.
Answer
(a) Expected value = β\0.04 per game. (b) The law of large numbers guarantees that the casino's average profit per game approaches \0.04 as the number of games grows, making long-run profit virtually certain.
The law of large numbers is the mathematical foundation of the gambling industry. While short-run outcomes are unpredictable, long-run averages converge to the expected value. A negative expected value for the player means the casino always wins in the long run.
About Law of Large Numbers
The Law of Large Numbers states that as the number of independent, identically distributed trials increases, the sample average converges to the theoretical expected value (population mean). In other words, larger samples produce more reliable estimates of the true probability or average.
Learn more about Law of Large Numbers βMore Law of Large Numbers Examples
Example 1 easy
A coin is flipped and the running proportion of heads is recorded: After 10 flips: 0.70, after 50: 0
Example 3 mediumA basketball player has a career free-throw percentage of 80%. She misses 5 free throws in a row in
Example 4 hardAn insurance company insures 100,000 homeowners. The probability of a claim in a given year is 0.02,