Odds of Winning Lottery: Expected Reward & Profit

[kelly0303]

Hello! My friend got me a lottery ticket (which I didn't win) and I decided to check the odds of winning for that particular game. The prizes for this game are: 5, 10, 15, 20, 50,100, 500,1000, 5000,1000000 ($) and the probability for each of the prizes is 1 over: 10, 10, 150, 50, 150, 131.63, 1636.36, 6545.45, 72000, 3276000. If the probability of winning a given price is p, then the probability of winning once by playing n times is: $p = 1-(1-p)^n$. So the expected reward is $\sum_i (prize_i \times 1-(1-p_i)^n)$. So I did the math and the expected reward is for increasing values of n, starting from 1: 3.9, 7.7, 11.3, 14.8, 18.1, and given that the price of the ticket is 5$ the expected profit is: -1.1, -2.3, -3.7, -5.2, -6.9. This means that the more tickets you buy, the less you are expected to gain. Am I doing something wrong, because this makes no sense. I expect that the more you buy a product, the more convenient it should become for you (for example the product discounts when you buy more at once). It's like buying something from a shop for 10$ and then every time you buy it again it's 5$ more expensive. Why would I buy more than 1 ticket? Thank you!

 

[suremarc]

Congratulations; you’ve discovered an essential truth behind gambling. It’s perfectly natural for the expected profit to be negative; that’s their entire business model. If the expected profit were positive, the ones funding the lottery would be losing money.

Also, it’s good that you’re curious about this, but the formula you gave is incorrect in several ways. The probability you gave only accounts for the case of winning 1 time, and it is also missing parentheses (though I believe you had them in your calculations). It will be approximately true for small values of $p_i,$ though.

Firstly, suppose the cost of a ticket is $C$, and consider the simple case where there is only one prize $X$. The expected profit from a single ticket is $Xp-C,$ where $p$ is the chance of winning. The outcome of each ticket is independent (unless you buy a stupid amount of tickets, in which case this is only approximately true), so the profit for buying $n$ tickets is just $n(pX-C).$

More formally, the number of wins after buying $n$ tickets follows a binomial distribution, which has expected value $np.$ Multiply that by the prize and subtract the costs to get the expected net profit.

In the case of multiple prizes, if each ticket nets you at most one prize, the correct formula to use is the multinomial distribution.

 

[Halc]

It's up front about being a losing proposition, even without taking into account the extra costs like taxes and the fact that the large prizes are often annuities, not cash.
No, I've never seen a lottery offering a volume discount. It's a tax on stupid people. Why discount that?

 

[Vanadium 50]

It's like buying something from a shop for 10$ and then every time you buy it again it's 5$ more expensive.

No, it's like buying something worth $5 from a shop that costs $10. Every time you buy one, you lose the same, not more or less.

 

[kelly0303]

No, it's like buying something worth $5 from a shop that costs $10. Every time you buy one, you lose the same, not more or less.

Right! I guess I expressed myself the wrong way. The main point is that, for some reason, I thought that the more you play the higher chances to make a profit you have (hence I thought this is why many people buy tickets every week). Of course the profit would always be negative, as the lottery wants to earn money, but I assumed that the magnitude of this negative number would at least get smaller the more you play. By checking these odds (and as you said in your example), the more I play the less profit I make. And this makes me wonder why would I (or anyone) buy lottery tickets at all, knowing that I would literally lose more, the more I play.

 

[PeroK]

Right! I guess I expressed myself the wrong way. The main point is that, for some reason, I thought that the more you play the higher chances to make a profit you have (hence I thought this is why many people buy tickets every week). Of course the profit would always be negative, as the lottery wants to earn money, but I assumed that the magnitude of this negative number would at least get smaller the more you play. By checking these odds (and as you said in your example), the more I play the less profit I make. And this makes me wonder why would I (or anyone) buy lottery tickets at all, knowing that I would literally lose more, the more I play.

Imagine a scenario where a) you feel there is no hope of improving your personal circumstances and are condemned to a life of relative poverty for you and your children; b) for a small weekly sum, you can buy a small chance of winning a large sum of money that would transform your life; c) this allows you better to accept your relative poverty in the hope of a rich life some day; d) you may even believe that it's inevitable you will win it one day.

Now mister the day the lottery I win I ain't ever going to ride in no used car again - Bruce Springsteen

 

[Vanadium 50]

you may even believe that it's inevitable you will win it one day.

Technically it is, provided you live long enough.

My thinking is that your odds of winning are the same to within six digits whether you buy a ticket or not.

 

[BWV]

Although the large jackpots in state lotteries fairly often have positive expected values - if there is no winner the pot grows for the next drawing. However, the odds remain so low, that its still not worth playing unless you can buy a full set of tickets, like this guy attempted:

https://www.washingtonpost.com/hist...ath-his-biggest-jackpot-still-came-down-luck/

 

[MathematicalPhysicist]

Congratulations; you’ve discovered an essential truth behind gambling. It’s perfectly natural for the expected

Right! I guess I expressed myself the wrong way. The main point is that, for some reason, I thought that the more you play the higher chances to make a profit you have (hence I thought this is why many people buy tickets every week). Of course the profit would always be negative, as the lottery wants to earn money, but I assumed that the magnitude of this negative number would at least get smaller the more you play. By checking these odds (and as you said in your example), the more I play the less profit I make. And this makes me wonder why would I (or anyone) buy lottery tickets at all, knowing that I would literally lose more, the more I play.

Life is a gamble, you lose some and then... it's game over.