The Meaning of Mathematical Expectation

[robert Ihnot]

Playing video poker, 9/6 Jacks or Better, has an expected return, computer calculated, of 99.54% with perfect play.

Many players today certainly regard mathematical facts as much more useful than a rabbit's foot. They want to know the odds. During a progressive game, where the value of the Royal Flush goes way up then EVERYBODY wants to play as the mathematical expectation has risen above 100%

BUT a Royal Flush comes about once every 40,000 hands, and at 500 hands/hour, it would take days for the average player to obtain one. So most players don't get one--does this change their mathematical expectation?

The same thing can be said of the lottery, with rollover the expectation might exceed the cost of the ticket, and millions buy tickets. Yet usually only one person wins, and the state lottery does not goes broke either. Is something wrong with how players interpret the concept of mathematical expectation?

 

[matt grime]

Lotteries don't go broke because they aren't paid out on expectations alone. Only a proportion of the ticket sales goes back to the prize fund.

A gambling company in the UK got round the problem that there is allowed to be only one "lottery" in this country by paying strictly on expectations, they aren't broke.

I can't quite decpiher exactly what you're asking to be honest, perhaps if you explained what the expected return is, since expectations are usually numbers, in this case earnings, or are you saying that the expected earnings is 99.54% of the cost of play.

 

[chronon]

The same thing can be said of the lottery, with rollover the expectation might exceed the cost of the ticket, and millions buy tickets. Yet usually only one person wins, and the state lottery does not goes broke either. Is something wrong with how players interpret the concept of mathematical expectation?

The rollover happens when no-one wins the jackpot in a given week - that means that the lottery gets to keep the money that would have been paid out as a jackpot, and so can add it to the total next week. So the expectation might exceed 100%. The extra money comes from those who played the previous week rather than from the lottery.

In practice the return on the lottery is considerably less than 100% (the state is using it to raise money), and with a rollover the extra players will dilute the expectation, so it is likely to drop below 100%.

 

[matt grime]

100% of what though, expectations aren't usually percentages unless given as a percentage of something. what is the something that everyone seems to inuitively know here?

 

[robert Ihnot]

100% of what though, expectations aren't usually percentages unless given as a percentage of something. what is the something that everyone seems to inuitively know here?

The expectation would be the return on, say, a $1.00 lotery ticket. Suppose this actually goes to $2.00, but only one person out of 75,000,000 wins the prize of $150,000,000.

Take the St. Petersburg paradox, how much should a person pay to play the game? Suppose the person is to toss a coin until first head appears. His prize will be 0 if a head on the first toss, 1 if on the second toss, 2 if on the third, and continuing to double in that manner until he throws a head. Then the expectation will be:

(1/2)x0 + (1/4)x1+ (1/8)x2 +(1/16)x4 ++++ = 1/4 + 1/4 + 1/4 +++=infinity. So the person should pay an infinite sum to play.

This does not seem quite right and indicates that as possibilites become greater and greater, the usefulness of mathematical expectation might be questioned. After all, if only one person out of 75,000,000 is going to win the lottery, what does the expectation of $2.00 for a $1.00 ticket really mean?

 

[matt grime]

I know what expectation is, that's why i asked the question seeing as it appeared no one else did as they gave unusual and ill-defined answers. (I have a real valued r.v., its expectation is 100%, is meaningless.)

Expectation is the mean (earnings in this case), if you don't like its utility you are free to use the modal or median values. There are bonuses to each that I'm sure you can find discussed in many places, mean has we well known 'meaning' and will become more relevant if, in this case, you play the game a large number of times. As people aren't playing the lottery a large number of times indeed it might not be the best indicator to a player, however, anyone who plays the lottery needs their head examined.

 

[robert Ihnot]

I know what expectation is, that's why i asked the question seeing as it appeared no one else did as they gave unusual and ill-defined answers. (I have a real valued r.v., its expectation is 100%, is meaningless.)

Expectation is the mean (earnings in this case), if you don't like its utility you are free to use the modal or median values. There are bonuses to each that I'm sure you can find discussed in many places, mean has we well known 'meaning' and will become more relevant if, in this case, you play the game a large number of times. As people aren't playing the lottery a large number of times indeed it might not be the best indicator to a player, however, anyone who plays the lottery needs their head examined.

I see what you mean there, thanks for the comments. The average return to the lottery player is about zero. Those high jackpot games have very little in secondary prizes.

 

[CRGreathouse]

I see what you mean there, thanks for the comments. The average return to the lottery player is about zero. Those high jackpot games have very little in secondary prizes.

With the lottery, I'd suggest calculating probability of winning the jackpot over one's lifetime at a given rate of buying rather than the return per ticket, since the payoff is so rare compared to the number of tickets a person might be expected to buy.

Average expected return is useful if you are likely to win repeatedly (say, a 1/10 chance event if you play 1000 times). It's not very telling for a lottery. Another reason expected return isn't that appropriate is that typical people aren't risk-neutral.