Positive net expectation values and gambling

[Dauphin]

Euromillions draw tomorrow:

Loosely speaking....
Prizes - €180,000,000
Ticket combinations - 76,275,360
Ticket cost - €2
Net expectation. + €0.40 per ticket.

I've got my tickets, have you?


Feel free to discuss the what you will do with the money, or the oversimplification of my above calculation.

[KrazyHorse]

The chances that you will have to share a prize with somebody else destroy your profit margin.

[Dauphin]

There is still more in prizes than in ticket purchasing costs.

I trust all you are arguing is don't go out and buy all the possible combinations?

[KrazyHorse]

Doesn't matter. Even if you bought one of every ticket you would be returned less than 180 million, on average

[KrazyHorse]

No, what I'm arguing is that your expected return is going to depend on the prize distribution structure and the number of other people that bought tickets

[Dauphin]

Originally posted by KrazyHorse
Doesn't matter. Even if you bought one of every ticket you would be returned less than 180 million, on average

If you individually did then yes, but across all people who purchase tickets they will get all the prize money with that expected return rate.

Assuming you aren't about to skew the stats by buying significant number of tickets (which for sure is a bad investment )and the total number of tickets sold is known pretty well then I don't see why expectation will change regardless of distribution.

[Japher]

here, if two people win, the splot is split between those two people... the more people who win the more the pot is split.

Is that not the case in Britland?

[Dauphin]

And if everyone won they would be up.

[Dauphin]

Actually, I see where I went wrong now in one of my assumptions - that tickets sold = possible combinations. Would be interesting find out the number of tickets sold.

[Ben Kenobi]

Dauphin, there are 76 million combinations.

Let's assume there are precisely 76 million people buying these tickets, and each person buys only one ticket.

If you assume a perfect distribution in tickets, each person has exactly the same odds to win the same amount of money.

But what if the distribution isn't perfect? Suppose one other person has your number, leaving one number unpicked and two people, you and the other person with the same number.

What are your chances of winning money? You can expect to only win half the amount the other folks would.

Also, you now have the chance, not only to split the pot, but also that no prize is awarded. The worse the distribution, the worse the chances of each individual to win money.

[KrazyHorse]

a) Buying all the tickets isn't a bad investment. In fact, if buying 1 ticket is a good investment then buying all the tickets is a better investment (assuming that you buying tickets does not cause other people to buy more tickets). Your expected return is higher per ticket and your variance is lower if you buy all the tickets.

b) The distribution of prizes (specifically the distribution between fixed prizes and shared pool prizes) and the number of other people buying tickets is absolutely critical to determining the expected return on investment.

Both a and b can be seen by imagining the following scenario: say there is already 180 million EUR in the prize pool due sales from previous weeks. This is not true, because the 180 million figure only represents the jackpot, and in addition because the 180 million includes estimated sales from the current week, but let us assume this for simplicity. Now say that the lottery uses 60% of ticket sales to augment the prize pool. Imagine a few different scenarios, and all the money is in the jackpot:

1) Nobody buy you buys any tickets and you buy one ticket
2) Nobody but you buys any tickets and you buy all the tickets
3) 1 billion other tickets are sold and you buy one ticket

In situation 1 the prize pool is 180 million plus 1.2 EUR. Your expected return is the 2.4EUR that you quoted earlier

In situation 2 the prize pool is 271.5 million. Your expected return is 3.56 EUR per ticket, and there is no chance in the game

In situation 3 the prize pool is 1380 million. Your expected return is going to be slightly less than 1 billionth of this (in fact it will be 1 billionth times the chance that somebody in that 1 billion wins, which is close to 1, given the large number of purchased tickets). So your return is less than 2. In fact it's less than 1.38 EUR

[Dauphin]

BK. You are applying a person specific condition, I'm talking about a random punter buying a ticket.

In such a case, my personal drop in expectation is matched by everyone else's increase in expectation - this is because any main jackpot prize money goes to the next tier of prizes. (Technically, not this draw but the next one, as there is a limit of 12 rollovers)

[Dauphin]

Originally posted by KrazyHorse
a) Buying all the tickets isn't a bad investment. In fact, if buying 1 ticket is a good investment then buying all the tickets is a better investment (assuming that you buying tickets does not cause other people to buy more tickets). Your expected return is higher per ticket and your variance is lower if you buy all the tickets.

b) The distribution of prizes (specifically the distribution between fixed prizes and shared pool prizes) and the number of other people buying tickets is absolutely critical to determining the expected return on investment.

Both a and b can be seen by imagining the following scenario: say there is already 180 million EUR in the prize pool due sales from previous weeks. This is not true, because the 180 million figure only represents the jackpot, and in addition because the 180 million includes estimated sales from the current week, but let us assume this for simplicity. Now say that the lottery uses 60% of ticket sales to augment the prize pool. Imagine a few different scenarios, and all the money is in the jackpot:

1) Nobody buy you buys any tickets and you buy one ticket
2) Nobody but you buys any tickets and you buy all the tickets
3) 1 billion other tickets are sold and you buy one ticket

In situation 1 the prize pool is 180 million plus 1.2 EUR. Your expected return is the 2.4EUR that you quoted earlier

In situation 2 the prize pool is 271.5 million. Your expected return is 3.56 EUR per ticket, and there is no chance in the game

In situation 3 the prize pool is 1380 million. Your expected return is going to be slightly less than 1 billionth of this (in fact it will be 1 billionth times the chance that somebody in that 1 billion wins, which is close to 1, given the large number of purchased tickets). So your return is less than 2. In fact it's less than 1.38 EUR

You are absolutely right, of course. The scenario that exists is x number of tickets are bought and I buy 1 ticket. What will x have to be for me to have a net zero expectation? That'll be the key figure won't it?

[Dauphin]

Or assuming you know x, how many tickets should I buy to maximise expectation?

[Japher]

at least 1