Originally posted by Rusty Thompson:
The hype over the dollar amount is funny. Goes right along with the presentation by Bruce Norris on what drives the economy of real estate. For some reason it is a better idea to buy a ticket when the amount is high than it is when the amount is low. Nothing has changed, the odds never change. But people get hyped up so everyone has to go out and buy a ticket because they think they may be missing out on something.
Sorry Rusty Thompson, but the amount you can win absolutely does change whether or not its a good bet. In any betting situation, the odds and the potential win amount are multiplied together to determine a "expectation value". If you were betting a dollar on a coin flip and you win $2 if you guess correctly, the expectation value for your $1 bet would be $1.
Probability of losing 0.50 (i.e., 50%), amount won $0 (i.e., you lose your $1) = $0
Probability of winning 0.50, amount won $2 = $1.00
total = $0 + $1.00 = $1.00
That's called "true odds", which means the game pays out the same as the odds of winning.
You almost never actually have that in any casino game. The one exception is the odds bet in craps, which really does pay out true odds. Most games have a house edge. To put a edge into this coin flip, we could make it pay out only, say, $1.90 if you win. Now it looks like:
Lose probability 0.50, return $0 = $0
Win probability 0.50, return $1.90 = $0.95
Expectation value = $0 + $0.95 = $0.95
That's equivalent to roulette where the wheel has 0 and 00 (common in the US). The odds of hitting a number are 1 in 38 but they only pay you 1 in 36. Play long enough and your money disappears.
Now, if the coin flip game paid out $2.10, the outcome would be quite different:
Lose probability 0.50, return $0 = $0
Win probability 0.50, return $2.10 = $1.05
Expectation value = $0 + $1.05 = $1.05
If you could play this game, and you had enough bankroll to survive the inevitable losing streak, you could play forever and your pile of money would just get bigger. There are video poker machines in Vegas that are positive expectation. Trouble is, the denomination is low and the net return works out to about minimum wage.
On Tuesday, the cash payout on megamillions was $260 million. After taxes thats about $156 million. So you get:
Lose probability effectively 1.0 return $0 = $0
Win probability 1 in 176 million, return 156 million = $0.886
Expectation value = $0 + $0.886 = $0.886
Negative expectation game
With today's estimated cash payout of $462 million, giving you $277 million after taxes, the picture is much different
Lose probability effectively 1.0 return $0 = $0
Win probability 1 in 176 million, return 277 million = $1.57
Expectation value = $0 + $1.57 = $1.57
A positive expectation game. In fact, and incredibly positive expectation game. If there was a machine in Vegas that had that high of a payout, people would be killing each other to get at it.
For megamillions, the break even number is about $406 million. If you take the cash and pay the taxes, you're left with right at $176 million, giving you true odds on your bet. As soon as it didn't hit on Tuesday, and the projected jackpot went over $400 million, it became a whole new ballgame. And that's why its now rocketed up from $260 million on Tuesday to $640 million today.
You are correct that you will most likely lose because of the very long odds. But the payback is good enough that its a good bet to make.