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## Batting .400 and the Law of Large Numbers

Rod Carew, one of the few to make a serious run at .400 since Williams, has studied the .406 season and contends that Williams’s absences were a blessing.

“The fewer at-bats any hitter has over the required number of plate appearances, the better his chance is of hitting over .400,” Carew wrote in an e-mail responding to questions about 1941. “When I hit .388 in 1977, I had 694 plate appearances and 616 at-bats (239 hits). Ted had something like 450 at-bats in 1941 when he hit .406, and I think George Brett and Tony Gwynn had fewer then 450 at-bats when they made their runs at .400.

“All in all, the less at-bats, the better.”

He’s trying to articulate the Law of Large Numbers.  Anyone hitting near .400 is deviating from the expected proportion of hits.  If you flip a coin 10 times, you just might get 7 tails.  If you flip if 1000 times, there’s no chance you’ll ever get 700 tails.  Many people may bat .400 during a single game (a handful of at-bats), but almost no one does as the number of at-bats increases.  Their average converges to the more realistic season average.

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## Why Casinos Don’t Lose Money

So, how does this tie into casinos?  Let’s take the roulette wheel as our example.  There are 37 total numbers.  18 reds, 18 blacks, and 1 green.  If you guess red or black correctly, you’ll get a 1:1 payout (ie: If you bet $1, you’ll get back$2, thereby winning $1). If the wheel lands on green (0), both red and black lose. This is where the casino gets it’s edge in this particular gamble. Let’s calculate the expected value of a$1 bet on red.
$$E[X] = \frac{18}{37}(\1)+\frac{18}{37}(-\1)+\frac{1}{37}(\-1) = -\.03$$
What this means is you have an 18/37 chance of winning $1 (if it lands on red), and 18/37 chance of losing$1 (if it lands on black), and a 1/37 chance of losing $1 (if it lands on green) The expected profit for playing this game is negative 2 cents. Now, sometimes you’ll win, and sometimes you’ll lose, but if you play enough times, you’ll be averaging a loss of 23 cents per round. This is where the law of large numbers comes into play. As long as enough people are playing, the house will be averaging a profit of 2 cents for every dollar bet on that roulette table. Question: What is the expected value of correctly guessing a specific number? There are 37 numbers, but the payout is 35:1 (You get paid$35 for each dollar you bet)  Based on this answer, is it smarter to try guessing the color or guessing the number?