Why do the Expected Values differ, for different wagers of the same lottery?
For all the different wagers of Daily Keno's $10 PICK, the odds of winning jackpot are alike : 1 in 2,147,181. Then I calculated their EV.
| Wager | Jackpot | EV $= [(Jackpot - Wager) \times Probability] - Wager$ |
|---|---|---|
| $10 | $2.5m | $\frac{2.5E6 - 10}{2147181} - 10 = -8.84$ |
| 5 | $1.25m | $-4.42$ |
| 2 | $500k | $-1.77$ |
| 1 | $250k | $-0.88$ |
I know that lottery players can be irrational. But why do EV's differ by wager? These mighty differences in EV feel illogical, because any rational player shall solely play the wager with the highest EV (the $1 wager in this case).
Or have I overlooked reasons to play the other wagers with lower EV?
1 answer
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| Chgg Clou | (no comment) | Jul 18, 2023 at 06:55 |
TL;DR They are all the exact same losing bet, on average 88.4% loss
The key is that the EV is calculated in dollars, which then should be compared to the wager, which is also in dollars.
Bet \$1, and you have an expected loss of \$0.88. Bet \$2, and you have an expected loss of \$1.77 (2 x \$0.88) Bet \$5, and you have an expected loss of \$4.42 (5 x \$0.88) Bet \$10, and you have an expected loss of \$8.84 (10 x $0.88)
The house always wins. In fact, the odds are really high for the house, but that is understandable because the nature of typical government run lotteries is that they have to:
- Pay substantial prizes to entice the players (see more below)
- Pay the retailer a commission
- Pay for expenses (the printed lottery tickets, machines for printing the tickets and displaying the winning numbers, etc.)
- Provide a substantial profit for the government to use towards the stated programs (education or health or transportation or whatever - though I would argue it doesn't really matter - money is fungible and it all, in the end, becomes a voluntary tax that is used by the government as it sees fit).
So why in the world would someone take such bad odds?
- Entertainment. That is apparently a big part of Keno. Not my thing, but some people will sit for a long time in a bar or restaurant playing Keno.
- The big win. This is very subjective. Arguably, \$2.5 million is a lot of money for the vast majority of people. A quick Google search says that \$2.5 million net worth would put you in the top 2% of the US. Is it worth \$10 for a 1 in 2 million chance at \$2.5 million? For some people apparently it is.
- The relatively-big win. This is where I would expect to see the \$1 bets. \$250,000 is not \$2.5 million, but for a lot of people it would make a big difference in their lives, and \$1 is close enough to $0 for a lot of people to try that type of bet.
In any case, however the players may justify spending their money, whether they choose \$1, \$2, \$5 or \$10 doesn't change the odds and value. And since most of the time they are going to lose (though many games will provide smaller payouts for partial matches of the winning numbers, so it isn't necessarily as simple as "1 in 2 million") it really comes down to how much they feel like throwing away for a tiny chance at big win.
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