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The lottery paradox

There is a lottery with 1,000 players, each with one ticket that contains one number from 1-1,000. There is exactly one of each number on the tickets. One of the numbers is drawn and the player with that number wins. What are the chances that one of the players will win the lottery? The answer is 1 in 1,000. That is, it is highly unlikely that any player will win. Why? Well, go through each player's chances and you will see. Player 1 almost surely won't win. Neither will play 2. And so on all the way through player 1,000. Therefore, it is highly unlikely that any player will win. Am I right?

Lolwut? I do hope you're not serious about this, because you'd need to be mathematically illiterate to be so.

Probability is a measure on sets, not on individual elements. If the measure on the set is 1000 then the measure on each element regardless of the order in which it is mentioned is 1/1000. In short each of the tickets acquires the probability 1/1000 at the same time.

And maybe the first ticket was the winner and the others didn't have a chance. Highly unlikely does not mean NO chance.

ruveyn

No, you aren't. The probability that any specifically person would when is 1/1000, but for the total set it's 1. If you go through each person's chances, the probability of anyone winning the lottery would be calculated by person 1

*or*person 2 winning. As two different people winning is mutually exclusive, this is simple enough to calculate, just add the probabilities up. 1000 * 1/1000 = 1. Even if we go through your warped logic where it is possible for no one to win, the chance isn't 1/1000. If the probability of individual people winning is independent of the others than using De Morgan's law and the knowledge that the probability of A

*and*B is A * B, then the result you should get is 0.63230457522, not 0.001.

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sliqua-jcooter

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The lottery paradox isn't really about statistics, it's about logic. It is logical to assume that each individual ticket will not win, therefore it is logical to assume that none of the tickets will win.

However, it is also logical to assume that one of the tickets will win - thus the basis for the paradox.

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No, it's logical to assume that one out 1000 tickets will win.

none of the tickets winning nullifies it being any form of a lottery.

It's like saying 0.001 = 0, which clearly is not true.

As stated before, remote odds does not mean zero chance.

The "lottery paradox" isn't even a paradox, it's just flawed reasoning.

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sliqua-jcooter

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You're mistaking formal logic and informal logic.

That's the definition of a paradox - a logic argument that contradicts itself and still evaluates to true obviously employs a flaw in reasoning somewhere to arrive at the answer.

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You're also describing a "raffle" and not a "lottery."

A raffle must have a winner. 1,000 tickets and one will be drawn as the winner.

A lottery does not have to have a winner. 1,000 tickets but the winning "number" is generated at random and may not match the number on any of those 1,000 tickets.

That's why some states used to allow raffles as legal but prohibited "lotteries" as gambling.

**a**person winning the lottery, in that cenario, is 100%. The probability of a

**specific person**winning the lottery is 0.1%

Then you are doing conditional probability. The conditional probability of ticket 1000 winning give that 1 thru 999 did not win is 1.

ruveyn

**a**person winning the lottery, in that cenario, is 100%. The probability of a

**specific person**winning the lottery is 0.1%

Then you are doing conditional probability. The conditional probability of ticket 1000 winning give that 1 thru 999 did not win is 1.

ruveyn

I don't get it.

# of people who can win the lottery: 1000

# of possible lottery numbers: 1000

Therefore, the probability would be 1.

Probability is sometimes very straightforward and sometimes very tricky. Am I missing something?

**a**person winning the lottery, in that cenario, is 100%. The probability of a

**specific person**winning the lottery is 0.1%

Then you are doing conditional probability. The conditional probability of ticket 1000 winning give that 1 thru 999 did not win is 1.

ruveyn

I don't get it.

# of people who can win the lottery: 1000

# of possible lottery numbers: 1000

Therefore, the probability would be 1.

Probability is sometimes very straightforward and sometimes very tricky. Am I missing something?

If we are looking for the probability that one particular person among the ticket holders will win, then it is 0.1%, as stated above. If you're looking for the probability that there will be a winner among the 1,000 holders of the 1,000 tickets, then it is 1, which is the sum of the probabilities (i.e.: 0.1%) of all the individual holders. This is just a restatement of what ModusPonens wrote.

If you're looking at conditional probability, then you simply need to know who among the 1,000 ticket holders won, as Ruveyn explained.

One issue here is whether or not conditional probability applies. As stated in the opening of the thread, there's no need for Bayesian statistics; you're not conditioning on anything.

The other is the nature of probability. It is a measure of likelihood over a given set. The total likelihood is 1, meaning that someone wins. Given the size of the set, the likelihood of any particular individual winning is, of course, not high. So the probability of any one individual is 0.1, and that gives a total probability of someone winning as 1, as desired.

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