explain the law of large numbers when a gambler makes very many bets on red playing roulette

Because the probability of winnig $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 may bets on red.

I think the answer is, "He will go broke."

Quote: mtheobald5811

Because the probability of winnig $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 may bets on red.

How many is "very many"? That bet for $2 has an expected value of about -$.26, and a standard deviation of $1.997. With a relatively high house edge and a very low variance, your chances of coming out ahead are not very good. Some figures for different numbers of bets:

60 bets: ev -$6.32, SD $15.47
120 bets: ev -$12.63, SD $21.88
240 bets: ev -$25.26, SD $30.94
500 bets: ev -$52.63, SD $44.66
1000 bets: ev -$105.26, SD $63.16
5000 bets: ev -$526.32, SD $141.23

So, what does that all mean? The probability of coming out ahead for a given number of bets is dependent on the size of the expected loss relative to the standard deviation. In other words, how lucky do you have to be to win; how much better than the average expectation is required to overcome the "handicap" of the house edge. Note that the expected loss goes up with the number of bets, but the standard deviation (measure of variability) goes up with the square root of the number of bets, so the expected loss grows much faster.

If the expected loss is equal to the standard deviation (somewhere between 240 and 500 bets), you have to come out one standard deviation better than expectation to break even. The probability of that is about .16. As the number of bets increases beyond, it takes more and more luck to remain ahead. At 5000 bets, the expected loss is 3.7 times the standard deviation, and the probability is breaking even or better is less than .0002, or odds of greater than 5000 to 1 against.

Compare these figures to the pass line in craps for $5. After 5000 bets, the expected loss and the standard deviation are just about equal, so that player still has that 16% chance of breaking even or better.
Cheers,
Alan Shank

He is a gambler playing a negative expectation game.
He is buying HOPE.
He hopes he gets a lot of free drinks.
He hopes the cocktail waitress is stunning.
He hopes the dealer makes a few mistakes in his favor.
He hopes that he hits a string of reds as part of the natural variation in the expected results.
After he has accumulated a really impressive stack of chips, he hopes some really stacked broad will come up to him and say that she has noticed him and that he is ahead due to variance but that before the Law of Large Numbers catches up with him, she would like to draw his attention to large number of inches in her bra size, her low cut gown and other charms and she would like to invite him to leave the table a winner and come up to her palatial suite where she will reward him with a Banana Chip every time he makes her happy.

Those are his hopes.
What will happen is that he is just as likely to have that variance or standard deviation thingie go in the direction of more than the expected number of blacks.
The drinks will be free, but the waitress won't be all that stunning and drinks will be watered down a bit.
The dealers are just as likely to make an error against him as in his favor.
That stunning girl will be on someone else's arm and she will not be paying any man for sex.
He will slowly go broke because he is enjoying himself and therefore has no reason to leave the tables.
So over time, he will approach that expected result of taking the short end of the stick.

Quote: FleaStiff

Those are his hopes.
What will happen is that he is just as likely to have that variance or standard deviation thingie go in the direction of more than the expected number of blacks.

That is absolutely correct.
No matter what the game or the bet, variance works both ways. However, without it, there would be no possibility of winning.
Cheers,
Alan Shank

Quote: mtheobald5811

Because the probability of winnig $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 may bets on red.

The above were good answers. To boil it down to basics, the Law of Large Numbers says that over a very large number of trials, the average of the results will be very close to the expected results. To use roulette as an example, if you bet red millions of times, your ratio of wins should be very close to 18/38.

Quote: Wizard

...if you bet red millions of times, your ratio of wins should be very close to 18/38.

To expand on this, if you start with $1mil, and make 1 million bets on red, and put your winnings in a separate pile, when the $1mil is done, you should have about $947,386 in the other pile.

The problem is, if he's a typical gambler, he'll bet another 947,386 times, and will probably end up with about $897K. And then he'll do it again, and end up with about $850K. And then do it again. Etc.

The other problem is, gamblers typically do not keep the money in two piles.


So, bottom line, the first response, "He'll go broke" is probably the best answer.

But if you're tutti, the fact that you didn't get the exact number of reds in the pile after 1,000,000 bets means that the HA is bull.