How to compare lotteries, when one has highest probability of winning the jackpot, but another the highest Expected Value?
Postulate that I shall always pick the lump sum for Daily Grand. The data below showcases that Pr(Keno's jackpot) > Pr(Grand's jackpot) > Pr(Lotto 649's jackpot). $\color{red}{\text{This probability inequality tips you to buy Keno!}}$
But the Expected Values of
1 Lotto 649 play= $= \dfrac{$5E6}{13,983,816} - \$3 = -2.64$.
1 Daily Grand play = $= \dfrac{$7E6}{13,348,188} - \$3 = -2.48$.
1 Daily Keno play $= \dfrac{$2.5E6}{2,147,181} - \$10 = -8.84$.
Hence, E(Keno) < E(649) < E(Grand) < 0. $\color{red}{\text{This Expected Value inequality tips you to buy Grand!}}$ Plainly, the results from the Probability and Expected Value clash!
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Why does Keno have the highest probability, but Daily Grand the highest Expected Value? How do I construe this contradictory information? My mind is freezing from this cognitive dissonance!
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Presuppose a player can live snugly on any 1 of these lotteries' jackpot, and fancies winning whichever one. Then how ought this player construe the above computations? How ought this player decide which lottery to buy?
Data for 3 lotteries from the Ontario Lottery and Gaming Corporation
| LOTTO 649 | DAILY GRAND | DAILY KENO | |
|---|---|---|---|
| Jackpot | $5 million |
|
$2.5 million |
| Odds of winning jackpot | $\dfrac1{13,983,816}$ | $\dfrac1{13,348,188}$ | $\dfrac1{2,147,181}$ |
| Matching numbers required to win jackpot | 6/6 | 5/5 + Grand Number | 10/20 on a $10 bet |
| Number pool | 49 | 49 + Grand Number (7) | 70 |
| Number of tickets one person can buy | 10 | 5 | 2 |
| Price per ticket | $3 | $3 | $10 |
1 answer
To understand what is happening, consider smaller and simpler numbers.
Consider a first lottery with 1/100 chance of winning 10 units, and thereby an expected value of 1/10.
Then consider a second lottery with 1/1000 chance of winning 100 units. The expected value is still 1/10, even though the chance of winning has been divided by ten.
To make the second lottery more profitable on average, suppose the victory sum is instead 200. Now it has an expect winnings of 1/5, but the chance of winning is still 1/1000, which is less than 1/100 in the first game.
As for question number two, it sounds like you already know the answer. If any of the big rewards is sufficient, and you only have a single try, and if you have nothing better to do with the money (suppose the mafia will come at you unless you pay them a fantastic sum next week, for example), then, sure, go for the one with highest chance of winning a sufficient sum.
In a more complicated situation you will have to similarly figure out what your goal is and then calculate the probability of reaching that goal with various betting strategies. In particular, if you are going to be betting many times in a row, the expected value will become a reasonable proxy for many purposes.
But note that stock markets have had a historical positive return of 6 %, so if you are thinking of using a lot of money on betting, maybe consider some index fund with low running costs. Or a bank account; those are more profitable than lottery in the long run.
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