Probability of winning scratch off lottery from a store that just sold a winner

[asechrest]

This is an attempt to settle a dinner table dispute. The background is that it just made the news that a guy won $1M in a scratch off game at the gas station at the end of my road. My brother then remarked: "Well, we know which station NOT to buy from. What are the chances of two winners at the same store?"

The broad question is this:

Is the probability of my next scratch off purchase being a winner reduced by choosing to purchase the ticket from the same store that just won?

My contention is that my chances are not affected by this fact; that it's an independent event. I've attempted to explain this with a simplified scenario, but am not having great success.

The idea is not fully fleshed out, so you can make some reasonable assumptions. Finite number of tickets; multiple, but small number, of jackpot winning tickets; finite number of stores, etc.

So, who's right? A small bet is on the line! Thanks!

 

[SW VandeCarr]

My contention is that my chances are not affected by this fact; that it's an independent event. I've attempted to explain this with a simplified scenario, but am not having great success.

So, who's right? A small bet is on the line! Thanks!

For a fair lottery, the assumption has to be that this is an independent event.

 

[asechrest]

This was my attempt at simplifying the scenario to prove my point. Have I gone wrong anywhere?:

Lottery:

The question (I think), is whether the fact that a store just sold a winning ticket effects your individual chances of winning if you buy from that same store. So, let's take an extremely simplified example.

Setup:

• There are four total scratch off tickets
• Two of the four tickets are jackpot winners, the other two are non-winners

Q: What is the probability of a single ticket being a winner?
A: 2 of 4 tickets are winners, so there's a 2/4 chance, simplified to a 1/2 (50%) chance of pulling a winner.

Q: If you know one winning ticket has been removed from the pool, what is the new chance of a single ticket being a winner?
A: Since one winning ticket has been removed, we're left with a pool of three tickets, one of which is a winner. So the chance is 1 in 3, or 1/3.

New Setup:

• There are still four total tickets, with two winners
• There are now two stores, store A and store B
• There are two tickets in each store

Q: What is the probability that any single ticket is a winner?
A: 2 of 4 tickets are a winner, so your probability is still 2 of 4, or 1/2 (50%)

Q: Let's say that you know a winning ticket has been pulled from store A. What is the probability of you winning if you choose the remaining ticket in store A?
A: There are three remaining tickets (one in store A, two in store B), one of which is a winner (since the second winner has been removed from the pool). The probability that the remaining ticket in store A is the winner is 1 in 3, or 1/3.

Q: Again, let's say that you know a winning ticket has been pulled from store A. What is the probability of you winning if you choose your next ticket from store B?
A: There are three remaining tickets (one in store A, two in store B), one of which is a winner (since the second winner has been removed from the pool). The probability that you next draw from store B is 1 in 3, or 1/3.

Whether we choose to draw our next ticket from store A or store B has no bearing on the probability that we will choose a winner on our next ticket.

 

[SW VandeCarr]

There are many ways to frame these kinds of argument. The basic fact is that the outcome of one event does not affect the outcome of another event if the events are independent. In the case of a lottery, the winning tickets are presumably determined after the tickets have been distributed and sold. Therefore it's difficult to see how the fact that a particular store sold a winning ticket affects the probability of that store selling a winning ticket in the next lottery.

The argument is often made that if you toss a fair coin say 10 times and get heads every time, the odds favor getting tails on the 11th toss. In fact, the probability of heads on every toss is 0.5. One would have to explain how the outcome of one coin toss could possibly affect the outcome of another coin toss.

 

[chiro]

There are many ways to frame these kinds of argument. The basic fact is that the outcome of one event does not affect the outcome of another event if the events are independent. In the case of a lottery, the winning tickets are presumably determined after the tickets have been distributed and sold. Therefore it's difficult to see how the fact that a particular store sold a winning ticket affects the probability of that store selling a winning ticket in the next lottery.

The argument is often made that if you toss a fair coin say 10 times and get heads every time, the odds favor getting tails on the 11th toss. In fact, the probability of heads on every toss is 0.5. One would have to explain how the outcome of one coin toss could possibly affect the outcome of another coin toss.

There is a saying that "assumptions are the mother of all ****ups".

It seems a fair thing to assume independence and that the randomness principle is being applied to the stores (i.e. all places where tickets are drawn have the same chance of having a winning ticket), but to be honest it is always worth investigating these kinds of things to see if that is the case.

Chances are the distribution process for tickets might not be unbiased in the way that we assume. Even the process that creates winning tickets (which is most likely based on pseudo-random number generation) is probably going to have some kind of bias as well.

This also reminds me of the thread discussing the difference between underlying probability and likelihood, and your point really emphasizes that.

To the OP, I recommend you investigate the difference between underlying probability and likelihood to understand the ways that statisticians and experimenters analyze these kinds of problems.

 

[Stephen Tashi]

asechrest,

I suggest that you take your stake in the bet and buy lottery tickets at a store that has recently sold a newsworthy winner and that your friend take his stake and buy lottery tickets at stores that haven't recently sold a newsworthy winner.

 

[SW VandeCarr]

Probability models, like most models, depend on certain assumptions. So the assumptions regarding lotteries can be generally stated that the lottery is fair. I don't know the mechanics of the lottery operation, but to be fair, I would say no one can possibly know the winning ticket numbers until ticket sales are closed because the winning ticket numbers are only drawn at that time. Moreover, every available ticket number must be potentially available at every outlet. That is, outlets don't hold physical tickets, but print tickets from a central source at the time of sale.

If one gets only heads after 10 consecutive tosses of a presumably fair coin, the probability of such an event is P < .001. It's reasonable to infer, in this case, that the coin is not fair since the hypothesis that the coin is fair can be rejected at the significance level of P = .001, which is stronger than most scientific studies use. So fundamental assumptions can be questioned.

EDiT: In the real world, if you got such a result, you would repeat the experiment. In the first experiment, a fair coin would be an assumption. In the second experiment, a fair coin would be a hypothesis.