Link to Problem: HERE

A 1-dollar bet on craps has an expected winning of -.0141. What does the Law of Large Numbers say about your winnings if you make a large number of 1-dollar bets at the craps table? Does it assure you that your losses will be small? Does it assure you that if n is very large you will lose?

What does the Law of Large Numbers say about your winnings if you make a large number of 1-dollar bets at the craps table?

The law of large numbers says that as \(n \rightarrow \infty\) then your outcome will tend towards its theoretical value. That is, when \(n\) is small, the variation will be proportionally larger than if \(n\) were larger. Thus, the longer you play (or the more bets you make) the more likely that your outcome will match the expected outcome. Since the expected value is negative, this means that the longer you play, the more likely it is that you will end up down money, specifically \(-0.141\) times the number of bets in dollars.

Does it assure you that your losses will be small?

Quite the opposite. The larger \(n\) gets, the higher the likelyhood your loses will be larger.

Does it assure you that if n is very large you will lose?

I don’t know about assure, but it certainly makes it very likely. As \(n \rightarrow \infty\) it garuntees it.

I conducted a series of simulations consisting of four scenarios to determine how often you would come out ahead or behind after \(n\) bets:

  1. Betting n=11 times, repeated in 1000 “parallel universes”

  2. Betting n=101 times, repeated in 1000 “parallel universes”

  3. Betting n=1001 times, repeated in 1000 “parallel universes”

  4. Betting n=10001 times, repeated in 1000 “parallel universes”

Simulate.Roulette <- function(n){
  prob.win <- 18/37
  ahead <- 0
  for(i in 1:1000){
    ahead <- ahead + as.integer(sum(runif(n, 0, 1) <= prob.win) > n/2)
  }
  print(paste0("n: ", n, " Ahead: ", ahead, " Behind: ", (1000 - ahead), " Prop: ", (ahead/1000)))
}

Simulate.Roulette(11)
## [1] "n: 11 Ahead: 482 Behind: 518 Prop: 0.482"
Simulate.Roulette(101)
## [1] "n: 101 Ahead: 418 Behind: 582 Prop: 0.418"
Simulate.Roulette(1001)
## [1] "n: 1001 Ahead: 194 Behind: 806 Prop: 0.194"
Simulate.Roulette(10001)
## [1] "n: 10001 Ahead: 4 Behind: 996 Prop: 0.004"

Since the expected value is negative, we know that if you play long enough you will eventually lose all of your money. This is the law of large numbers. As the number of plays is small, the initially variation allows a greater likely hood that you may come out ahead. However as n grows larger, the likelyhood of winning will tend towards 0.

Conclusion: If you ever get to play Lebron James in a game of pickup you should play first basket wins as opposed to first to 11.