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tomgwyther

lottery paradox

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A man living near me won the lottery twice, both times he got five balls plus the bonus ball

The odds of his first win were 2,330,636 to one. But the second time, the odds shot up to 5,400,000, 000,000 to one. Yesterday, Mr McDermott celebrated aboard a yacht at the Port Solent marina near Portsmouth - symbolising the fact that he and his wife are soon heading off to the island of Kerkenah, off Tunisia, where they have bought a house.

 

According to this, winning the lottery once has odds of 2,330,636 : 1

Winning the lottery twice has odds of 5,400,000,000,000 : 1

Now lets assume that I won the lottery last week. If I buy a ticket for next weeks lottery draw, what are my odds of winning? 2.3 million to one or 5.4 trillion to one... they can't be both, can they?

 

Or, to put it another way. The odds on the same balls being drawn two weeks in a row must be staggering, but the likelihood of a particular sequence of six numbers being drawn on a particular day must remain the same. How does this paradox resolve?

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No, the odds do not “shoot up” the second time. The fact that he has already won the lottery before does not affect his chances of winning the lottery again in any way whatsoever. You are probably confused by two totally different questions:

 

(i) What is the probability of winning the lottery twice (rather than once)?

(ii) What is the probability of winning the lottery a second time (given that he has already won it once before)?

 

If the probability of winning the lottery in any one week is p, then the answer to (i) is p2 whereas the answer to (ii) is just p. It is important to be clear about the distinction between (i) and (ii).

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I thought as much. common sense and logic would tell me that my odds on winning the lottery are the same from week to week, regardless of what I'd won or hadn't won previously, as the lottery game has no memory

 

I'm still having trouble distinguishing between (i) & (ii) as both (although different in their time-line) involve winning twice.

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The odds would be the same. The occurrence was random. To make future predictions you need to compare the ratio of tickets you have to tickets sold. It would be the same no matter what happened to him.

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Think of it this way. The odds of winning the lottery twice in a row are 5.4 trillion:1. However, by already having won the lottery once, you've made a big dent in those odds, as all you have to do is win again, at a mere 2.3 million:1.

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Sisyphus has the right idea. The odds of winning twice are 5.4 trillion:1, but this assumes that you have not yet won. After the first game, your odds are 2.3 million:1, as the outcome of the first game has no affect on the second.

 

The reason the odds change is that before you win the first game, the odds of winning it are 2.3 million:1, but after you have won it, it is a certainty (i.e. 1:1) since it has already happened.

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