Games for Money

Let’s discuss the realities of expected value, if the lottery really is merely a “tax on the stupid,” and what it really means to gamble.

Most gambling situations have an expected win value. For instance, in a coin toss situation, if I bet $1 on heads to win $1, the expected value of my wager is exactly $1 - I have $2 if I win, and $0 if I lose, and there are only two states. This gets more complicated in situations where there is an element of house rake. Roulette, for instance, has two green squares such that even betting on black or red (which both pay the same $1 for a $1 bet) are not quite equitable in value because there are more chances to lose than there are to win. TO be specific, there are 18 black squares, 18 red squares, and 2 green squares, so that if you bet $1 on black, your equity is 18/20, or $0.90. That dollar you bet is only worth 90 cents once you’ve gambled it. The “true” equation for this is P₁xV₁ + P₂xV₂.

Put another way: if you bet on a horse with a 10% chance of winning, and the bet pays $100, the probability P₁ is 10% that your ticket is worth V₁ $100, and the probability P₂ is 90% that your ticket is worth zero. So your expected value would be:
0.1 x $100 + 0.9 x $0 = $10.
WHich means that if you went to the track and bet the horse 250 times, you would expect that 25 of those times would win, netting you $2500; if your bet was $10 each time your total wagering would also be $2500 for all 250 tries. If you pay less than $10 for the ticket, you are profitable, if you pay more than $10, you are NOT profitable.

So when considering money management or statistics, you consider your current equity in the proposition.

It’s exactly the same for lotto, and specifically in this example for Powerball.

Here is the completely WRONG train of thought:
According to the powerball website, winning the grand prize is 1 in 146,107,962.
so, using the formula, you’d think you should get
(1/146,107,962) x $340,000,000 + (146,107,961/146,107,962) x $0
or an expected value of $2.33 per ticket.

Absolutely FANTASTIC expected value. You just bought something worth roughly $2.33 for $1. Tell me again how that’s a bad thing.

This line of thinking is wrong because that you are gambling for your share of the powerball jackpot, because of the possibility of multiple winners. And that’s even assuming someone wins.

(actually, we can be pretty damn sure someone wins, because we can work that out. The rule of thumb for calculating the likelihood of a winner for 1 in x, with y chances of it happening is 1-e^(-y/x), which given the number of tickets sold results in a roughly 94% chance of picking a winner for the next draw)

So we can be pretty sure someone will win. The question is how many.

This results in a lovely bell curve, and I’ll spare you the statistical distribution, but the odds of nobody winning is 6%, of one winning is 22%, of 2 winning is 27%, 3 is 22%, 4 is 16%, and the odds of 5 or more winning is 7%. On average, the winner should expect a 38% return on the total jackpot, or $129,200,000. Which is still fan-freaking-tastic, as far as I’m concerned. But it does change our final expected value:

(1/146,107,962) x $129,200,000 + (146,107,961/146,107,962) x $0 = $0.88.
Dang. Less than $1.
Did we forget anything?
Yes.

You left out conditions where the ticket is a winner, but doesn’t win the jackpot. To make this post shorter and to save on your math headache, I’ll cheat and just tell you it adds 21 cents.

Grand total = $1.09 for each $1 ticket. Every $1 you spend on a powerball ticket gets you an extra 9 cents in value.

Many may scoff at this, and say “How ridiculous, lotteries are just a tax on teh stupid” as I’ve heard so many times, yet these are also the same people who are paying $6 for a hot dog at the baseball game. Why?

Expected value.

brother9 wrote this on October 20th, 2005 and filed it under Gambling |