Lottery probability question

[Darth_Panda]

@Askance

I tell you I have 3 coins in my pocket. What’s the probability that if I empty my pocket onto the floor, all the coins that drop will come up heads? 1/8 you’d say, no doubt. But I then discover there are in fact 4 coins in my pocket, so the true odds of all my coins coming up heads are 1/16. The extra information you gained had no effect on the actual probability, but rather allowed you to more correctly calculate what the odds really were. Even if I had said I had some coins in my pocket (so all you could say is the odds were 1/4 or lower), the fact remains the odds were precisely 1/16, regardless of your knowledge or lack thereof of what those odds were.

No, it’s not. That’s the probability as far as you are aware. Probability is a measure of the actual likelihood of an event, not the perceived or incorrectly calculated likelihood of it.

Yes, there is. If everyone in the world died tomorrow, and the day after that a shelf holding three coins collapsed, the odds those 3 coins come up all heads remains 1/8, despite no-one knowing about it.

Exactly - regardless of the state of your information about that situation.

In the lottery example, the winner has already been determined - the OP is now calculating the chance that he was the winner - which is different than calculating if he will be the winner prior to the drawing. Now, if after the drawing had occurred he still knew nothing about the drawing, then the odds of his having won would be the same as the odds that he would win when calculated before the drawing. However, as he continues to gain more and more information, the odds that he won change based on what he knows. For example, when he finds out that “Either you or Bill won” - the odds that he *did win *are now 1 in 2 (assuming they each had 1 ticket).

For your coin example - yes, the odds of all three coming up heads are 1 in 8 before they are tossed. After they have already been tossed, the odds they did land all heads up is 1/8 if we can’t see them. If we can that 1 is heads up, the odds are now 1/4 that all are heads, if we can that 2 are heads up then there is a 50/50 that all 3 are heads up.

[Darth_Panda]

Also, please read this:

http://www.straightdope.com/columns/read/916/on-lets-make-a-deal-you-pick-door-1-monty-opens-door-2-no-prize-do-you-stay-with-door-1-or-switch-to-3

“Why? A different example will make it clear. Suppose our task is to pick the ace of spades from a deck of cards. We select one card. The chance we got the right one is 1 in 52. Now the dealer takes the remaining 51 cards, looks at them, and turns over 50, none of which is the ace of spades. One card remains. Should you pick it? Of course. Why? Because (1) the chances were 51 in 52 that the ace was in the dealer’s stack, and (2) the dealer then systematically eliminated all (or most) of the wrong choices. The chances are overwhelming–51 out of 52, in fact–that the single remaining card is the ace of spades.”

"There is a family with two children. You have been told this family has a daughter. What are the odds they also have a son, assuming the biological odds of having a male or female child are equal? (Answer: 2/3.) …

The possible gender combinations for two children are:

(1) Child A is female and Child B is male.

(2) Child A is female and Child B is female.

(3) Child A is male and Child B is female.

(4) Child A is male and Child B is male.

We know one child is female, eliminating choice #4. In 2 of the remaining 3 cases, the female child’s sibling is male. QED. "

The Master

[CurtC]

@Askance

I tell you I have 3 coins in my pocket. What’s the probability that if I empty my pocket onto the floor, all the coins that drop will come up heads? 1/8 you’d say, no doubt. But I then discover there are in fact 4 coins in my pocket, so the true odds of all my coins coming up heads are 1/16. The extra information you gained had no effect on the actual probability, but rather allowed you to more correctly calculate what the odds really were. Even if I had said I had some coins in my pocket (so all you could say is the odds were 1/4 or lower), the fact remains the odds were precisely 1/16, regardless of your knowledge or lack thereof of what those odds were.

I disagree with this - in your example, if you say you have three coins and ask me what the probability is that all your coins will come up heads, I first have to to factor in an estimate of whether it’s correct that you have three coins. The probability will be somewhere around 1/8, but in a case where there’s room for you to make a mistake, I can’t figure the probability exactly unless I know the details of how accurate your original statement was.

No, it’s not. That’s the probability as far as you are aware. Probability is a measure of the actual likelihood of an event, not the perceived or incorrectly calculated likelihood of it.

That’s a good example that shows what I’m saying. In the OP case, the event has already occurred, so the way you describe it, there is no fractional probability - it’s either a winner or a loser, we just don’t know which yet. This is similar to the unknown number of coins in your pocket, estimated to be three. The probability that we’re calculating is based on the information we know at any given time, not on all the information available to anyone on the planet.

[don_t_ask]

I tell people a story about how, some years ago, I was on holiday with my family and read in the paper that someone had won the major prize in the lotto draw that week but no-one had come forward to claim it. I had a ticket in the draw but had left it at home and had no idea what numbers I had. I explained to my wife that it was now between “me and the other people who haven’t checked their tickets.” As the days went on, with the media haranguing people to “please dear god check your tickets,” I told my wife we were getting closer to being the winner, we were at least competing with less people each day. I would check the paper each day, to ensure no winner was found, and at breakfast estimate our chances. My wife thought it was pretty funny.

I was able to spin out this weird fantasy for 5 or 6 days, getting closer and closer to winning, and then the bastards printed a piece saying that the winner had still not come forward to claim the winnings from the ticket that had bought at the Easter Show; and it was all over, for I hadn’t bought mine there.

[Frylock]

@Chronos

To reiterate what Indistinguishable said: The information that there was a winner definitely improves your odds by some amount, but we can’t say how much it improves your odds without some sense of how likely it was for there to be a winner.

Just to clarify something for my own sake:

If you also found out that there was only one winner and there were more than 200,000,000 ticket purchasers, this would mean the probability that you’re the winner goes down, right? (Originally, the probability was one in 200,000,000, as stated in the OP. Now, with the new information that there was one winner and there were more than 200,000,000 who purchased tickets, the probability that you were that winner is less than one in 200,000,000, right?)

So finding out there was a winner improves your odds, but finding out there was a winner, and only one winner, and there were more than 200,000,000 who purchased tickets, would lower your odds, correct?

May seem basic, but I’m just making sure I haven’t misunderstood something…

[Frylock]

@Indistinguishable

Two things to keep in mind: A) You can pick any probability distribution you want and calculate the probabilities according to such a distribution. Anytime someone reports probabilities, it is with respect to some implicit distribution which is of interest for some reason (perhaps because it has suitable symmetry properties, perhaps because it corresponds suitably with some other data of interest, or so on).

Would be interested in seeing an illustration of what you mean, for those of us who haven’t looked at probability since High School.

[Frylock]

@Askance

I tell you I have 3 coins in my pocket. What’s the probability that if I empty my pocket onto the floor, all the coins that drop will come up heads? 1/8 you’d say, no doubt. But I then discover there are in fact 4 coins in my pocket, so the true odds of all my coins coming up heads are 1/16. The extra information you gained had no effect on the actual probability, but rather allowed you to more correctly calculate what the odds really were. Even if I had said I had some coins in my pocket (so all you could say is the odds were 1/4 or lower), the fact remains the odds were precisely 1/16, regardless of your knowledge or lack thereof of what those odds were.

If you take seriously the idea that probabilities measure of what you are calling “actual likelihood,” then the probability that they all come out heads is either 1 or 0. If they do come out all heads, the actual likelihood, it turns out, is 1. If they don’t, then the actual likelihood, it turns out, is 0.

Understanding probability as measuring “actual likelihood” (in the sense you’re using the phrase) doesn’t allow for anything to have any probability other than 0 or 1.

[Frylock]

@Frylock

Just to clarify something for my own sake:

If you also found out that there was only one winner and there were more than 200,000,000 ticket purchasers, this would mean the probability that you’re the winner goes down, right? (Originally, the probability was one in 200,000,000, as stated in the OP. Now, with the new information that there was one winner and there were more than 200,000,000 who purchased tickets, the probability that you were that winner is less than one in 200,000,000, right?)

So finding out there was a winner improves your odds, but finding out there was a winner, and only one winner, and there were more than 200,000,000 who purchased tickets, would lower your odds, correct?

May seem basic, but I’m just making sure I haven’t misunderstood something…

Ah… too late to edit. I’ve just realized that I forgot how lotteries work and ended up describing an impossible situation in the above!

I’d be right, though, if lotteries worked just by rolling a d200,000,000 each time someone buys a ticket and determining they are a winner if the die came up 1. Right?

[Telemark]

The 1 in 200,000,000 chance is that you’ve chosen the right number. It says nothing about other people choosing the right number as well.

[Frylock]

@Telemark

The 1 in 200,000,000 chance is that you’ve chosen the right number. It says nothing about other people choosing the right number as well.

Not sure if this is addressed to me or not, but FWIW, I knew that.

[panamajack]

@Frylock

Ah… too late to edit. I’ve just realized that I forgot how lotteries work and ended up describing an impossible situation in the above!

I’d be right, though, if lotteries worked just by rolling a d200,000,000 each time someone buys a ticket and determining they are a winner if the die came up 1. Right?

Regardless of how the lottery works, if there is exactly one winner, and more than 200,000,000 tickets sold, the probability that you are the winner must be less than 1/200,000,000[sup]*[/sup].

In particular, your alternate method would mean 1/200,000,000 before any information about the winner is announced, and reduced probability after (1/#tickets actually sold).
*This cannot be the case with the OP’s lottery, as Frylock pointed out already.

[Frylock]

@panamajack

*This cannot be the case with the OP’s lottery, as Frylock pointed out already.

I just came back to write that I’d changed my mind! I was doubly confused, I think.

Even in a lottery like that in the OP, you could still have just one winner while over 200,000,000 people bought tickets. For example, perhaps 400,000,000 selected the same losing combination, and just one person selected the winning combination.

[Tom_Tildrum]

Assuming that before the drawing, your odds are 1/(the total number of permutations) or 1/200,000,000. And to eliminate the multiple-winner issue, assume that if you pick the correct permutation, you are still a winner even if someone else does as well (maybe you split it).

After the drawing, if all you know is that someone won, is it not the case that your odds of winning are 1/(the total number of tickets purchased)? Down to a minimum of 1/200,000,000?

In other words, if 100,000,000 tickets were sold, your odds are now 1/100,000,000. If 200,000,000 were sold, your odds are coincidentally still 1/200,000,000 (they haven’t changed).

If 400,000,000 were sold, your odds cannot be lower than 1/200,000,000 (because that’s the maximum number of permutations), and may be higher, depending upon the extent of the overlap of permutations among those 400,000,000.

To sum up, if we don’t know the number of tickets sold, your odds might still be 1/200,000,000, but will almost certainly be higher, and in no case will be lower.

[panamajack]

I realized I didn’t think through the ‘impossibility’ I suggested about the OP’s lottery. It’s only impossible in the sense that there must be more than one winner for each multiple of 200 million unique ticket (thus far I don’t think anyone’s mentioned that if a lottery lets you choose your numbers, there may not necessarily be a winner if >200 million tickets are sold, but that’s a minor point).

But the probability of winning can decrease below 1/200 million once you know information about the outcome.

In perhaps clearer terms - every ticket has an equally likely chance to win . If 300 million played and only one person won, it can’t be possible for someone’s chances to be greater than anyone else’s. Therefore everyone must have a 1/300 million chance of holding the winning ticket.

(Reaching back to something Indistinguishable said - this is a uniform distribution; you could come up with a different distribution, but it would no longer be the case that every ticket has an equal chance, and you’d need to convince me why you’re more special than everyone else.)

edit: I tried to clarify ‘every ticket’, but if I missed it, assume that each person only buys one unique ticket.