Wikipedia:Reference desk/Archives/Mathematics/2019 January 26

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January 26

Powerball Paradox of Probability

In powerball lottery in USA.

The probability of ZERO pairs of adjacent numbers is 73.5% and

the probability of at least one pair of adjacent numbers is 26.5%

I have two candidates for powerball ticket

Candidate A is 2,7,10,14,15 and powerball is 17

Candidate B is 2,7,10,14,29 and powerball is 17

Unfortunately I only have money for a single powerball ticket.

therefore 73.5% of the time Candidate A has no chance of winning because it has one or more pairs of adjacent numbers (aka {14,15} )

therefore 26.5% of the time Candidate B has no chance of winning because it has zero pairs of adjacent numbers

Therefore I should spend my money on Candidate B when I buy my powerball ticket.

But the lottery officials tell me that the drawing of powerball is random therefore Candidate A has as much chance of winning as Candidate B

Now this is a paradox indeed! Ohanian (talk) 14:52, 26 January 2019 (UTC)

Ohanian, there is no paradox, 26.5% is simply the odds of picking any two consecutive numbers. That is independent of it being the winning pick. Roger (Dodger67) (talk) 18:01, 26 January 2019 (UTC)

By analogy, consider a random drawing which picks a single number from 1 through 3. Two thirds of the time the pick is odd (1 or 3) and one third of the time the pick is even (2). If you played odd/even, then odd is a more likely result, but that doesn't mean that 3 is more likely than 2. Likewise, if you could place a side bet that the winning Powerball draw will have zero adjacent numbers, then you would win that side bet 73.5% percent of the time. But that doesn't make one particular zero-adjacent-numbers ticket more likely to win than one particular at-least-one-pair-of-adjacent-numbers ticket; it just means that there are nearly three times as many possible tickets which fit that first description than there are those which fit the second description. -- ToE 02:03, 27 January 2019 (UTC) BTW, what you did here -- setting up a well defined problem which appears to be a paradox and asking for an explanation -- is an excellent learning technique. Good job!