Except that the conditions of the OP stipulate that you do not know how many people bought tickets or how many numbers were in play - without that information, your odds remain the same - 1 in 200,000,000. The ONLY thing that changes is that you know your odds are still 1 in 200,000,000 rather than 0.
Let’s say that you travel in time ahead to tomorrow, and catch the news to find out that “someone won the lottery” - but you don’t know who, how many people won, or what the winning ticket was. You travel back to today, and buy a lottery ticket.
You have no idea if anyone won or lost the lotto.
You find out that someone won.
You find out that someone in your state won.
You find out that someone in your county won.
You find out that someone in your city won.
You find out that someone in your neigborhood won.
You find out that someone on your block won.
You find out that someone in your house won.
Do you really think that the probability that you won is exactly the same in all of those scenarios… come on.
I know the population of my state, county, city, neighborhood, block and house. The OP does not know the population of “number of tickets sold”. Until we find that information out, the odds don’t change.
Consider 4 people in a room rolling a 10 sided die.
Before anyone rolls, each person has a 1/10 chance of rolling 1,000. Overall, there is a 34.39% chance that 1 person will roll a ten (1-0.9^4). You can actually draw this out with your traditional branching tree 1-1-1-2; 1-1-1-3, 1-1-1-4; …; 10-10-10-10. You will end up with a total of 10,000 possible branches, and 3,439 branches will include at least 1 ten.
If you travel into the future and find out that a ten was rolled, then you know that 1 out of the 3,439 possibilities must have occurred. Each of those 3,439 have an equal likelihood of having occurred. So you would simply calculate in how many of those scenarios you would have rolled a ten - it turns out that this is 1,000 (10^3). So, your odds would be 1,000 in 3,439; or 29.0782% - considerably better than 1 in 10.
You may know the population of your state, but you don’t know how many people in the state purchased a lotto ticket - so there is no functional difference between knowing that ‘someone’ won and knowing that someone in your state won. Are you claiming that if you know that someone on your block won, but you don’t know how many people on your block bought a ticket, then your odds are still the same as if you knew nothing?
This is exactly the right answer, but just in case the math is unconvincing, I ran some simulations. I assume that each ticket has a 1 in 100 chance of winning, that the tickets win independently, and that the number of tickets purchased in any given week is Poisson with mean 100. After one million weeks, the proportion of time that I win is 0.009872, but the proportion of times that I win conditioned on anybody winning is 0.01551791. Clearly, knowing that someone has won should cause me to increase my belief that I have won.
For those who are curious, here’s my R code:
rm(list = ls())
p <- 10^-2
lambda <- 1 / p
N <- 10^6
wins <- matrix(NA, nrow = N, ncol = 2)
for (i in 1:N)
{
n <- rpois(1, lambda)
plays <- runif(n + 1) < p
wins[i, ]<- c(any(plays), plays[n + 1])
}
prob.unconditional <- sum(wins[, 2])/N
prob.conditional <- sum(wins[, 2])/sum(wins[, 1])
Either way, my point still stands - in your example, you’re telling me exactly how many people bought lottery tickets. In the OP’s example, we don’t know that information.
I understand your point - and I certainly agree with ultrafilter’s proof. I think I’m just sticking to the fact that we don’t know any specifics at all, other than his odds are still not zero. If I knew someone on my block won, I can easily determine what my odds aren’t - i.e. better than 1 in 100, worse than 1 in 10.
The number of people rolling dies is irrelevant. If x amount of people in a room were rolling the die, you could solve the equation for every possible value of x, and the probability of rolling a ten would always be higher after obtaining the information from the future and, by induction, you could say with certainty that the probability of you rolling a ten is ALWAYS higher when you determine that someone has indeed rolled a ten. You could also inductively prove that as x approaced infinity then delta-prob(you) would approach zero - but it would never reach zero.
You are correct that you cannot determine what the new odds are without more information - but they are new odds and they are higher. The ‘block’ thing is more obvious because it is of a greater magnitude, but there is no qualitative difference between the greater probability given the block information and the someone information.
You can determine what your odds aren’t with the someone information - 1 way to lose is by having no one win - that loss is now off the table. And depending on how the lottery is set up, that could be a big deal or almost entirely irrelevant.
The OP clarified that most of the time, no one wins. He doesn’t know how many tickets were sold, but he can estimate based on how often someone wins.
I think this is why this is confusing. If you assume both “these lotteries go weeks between someone winning” and also “we have no information at all how many tickets were sold” you’re making contradictory assumptions. If lotteries go weeks between winners, then only a fraction of the possible numbers are sold each day. That’s information. It may not be precise, but it’s more than no information at all.
It seems to me there are two very different games being referred to as 'lotteries" here. To me a Lottery is a game where there are a fixed number of numbered tickets printed, all of which must be sold before the draw is conducted, and with a fixed prize amount. There will always be exactly one winner. The OP mentions a fixed (or at least maximum) number of tickets, but also that they may not all be sold in a given game - to me these are mutually exclusive. In this game knowing there’s a winner makes no difference whatever to your odds of being that winner, it just means all tickets have now been sold and the draw has then been conducted.
OTOH many people here are referring to what I call Lotto, a game where numbered balls are drawn and you bet on particular numbers coming up. These have unlimited subscriptions and the draw is conducted at a particular time, determined in advance, and there is not necessarily a winner as no-one may have chosen the combination drawn. The prize amount depends on the amount bet. In that case, yes, knowing there’s a winner does increase your odds of being that winner, since trivially if there was no winner you could not have won - any positive number is greater than zero. There’s no way of quantifying it further than that, on the information provided.
If we posit a game where a fixed number of numbered tickets are printed, some are sold, and a draw occurs at a set time regardless of sales, then we have a situation such as the OP describes (I believe), but that would be a dumb game since the lottery vendor could lose money (assuming a fixed prize). The odds of him being a winner once it’s been announced that there is a winner would still as described in my second para: an incalculable amount more than 0.
I guess I don’t see what wasn’t clear about my earlier post. I’ll try again. The probability of 1 in 200,000,000 can be broken down as the product of two other probabilities:
Let A = the probability that someone will win this lottery. In the OP’s description, this is significantly less than one.
Let B = the probability that, if someone wins the lottery, it will be me.
Now we already know that my ticket’s probability is
1/200,000,000 = A * B
We know that A<<1. What does that tell us about B?
It tells us that B, the probability, given that someone has won the lottery this time, that I am the winner, is way larger than one in 200,000,000.
The new information means that the probabilities have changed.
I think we’re talking about the lotteries that are run by the states, which you call by the name “Lotto,” which is a lottery game. If there are a fixed number of tickets, that would also be a lottery, but the state Lotto games are also lotteries.
Odds aren’t some thing that exist when you’re not looking at them. They’re just a measure of ignorance. They’re a way to quantify what you know about the possibilities. Otherwise, you get into the nonsense of ‘Your odds are either 0% or 100%, you just don’t know which.’
Of course the probability goes up. Of all the possible ways you could have lost, learning that “someone” won removes every combination of possibilities in which you and everyone lost. Some of the possible losing scenarios are impossible, meaning there is left over a greater ratio of winning scenarios to losing ones. Good luck doing the math, though.
Just to get started with the math, you’ll have to define a limit for the number of possible tickets sold. You don’t have to know how many tickets were sold, but you DO have to know how many it is possible to sell. You’ll also have to define whether we know exactly how many people chose the correct numbers, or if all the information we get is “at least one person won”. If we know “three people won” it changes your odds compared to “only one person won.”
If I may, I suggest “expected value” replace “probability”, in the sense that the expected value of the OP’s lottery ticket changes with each new piece of information. With no information, the ticket’s value (with some assumptions, like each ticket is unique) is (prize value * odds of winning).
If he knows nobody won, expected value = 0.
If he knows somebody won… well, then we need more information, like how many tickets were sold. In any case, if the OP was trying to sell his lottery ticket to another person, he can calculate an expected value based on the information at hand.
No, it means that our ability to correctly evaluate the probabilities has changed, improved in fact. The probability itself is unchanged by the mere fact that we have received new information.
What’s the difference between “our ability to correctly evaluate the probabilities” and the “probability itself”?
I mean, that’s what probability is - it’s a measure of the likelihood of an outcome based on the information that’s available to me. There’s not some Platonic probability that exists independently of the information that I know - in this situation, the ticket has already been chosen, so in reality my ticket either already is or already isn’t a winner.
The probability has improved when I find out there was a winner, because I have more information.
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.
