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I was looking over the PA lottery odds and payout table for the Pick 5 "Quinto" game, specifically for the second row "Boxed - 5 in any order - 5 distinct digits"

http://www.palottery.state.pa.us/Games/PICK-5/Prizes-Odds.aspx

where they list the odds as 1:833.33 and 120 winning combinations which works out to 0.12 %. What I am confused about is the math behind these odds..

For this game, you pick 5 distinct digits and can match them in any order to the 5 numbers drawn by the PA Lottery. There would be 252 ($_{ 10}C_{5}$) unique groups of five numbers that you can pick and 100,000 possible numbers that the PA Lottery can draw so shouldnt the probability be $\frac{252}{100000}$ or 0.252%?

celeriko
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1 Answers1

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Player picks $5$ different digits.

Lottery generates an ordered selection of $5$ digits (repeats are possible). Thus there are $100,000$ equally likely selections that the lottery might make.

In order for player to win, the lottery must pick the $5$ digits of the player in some order. There are $5!=120$ such ordered selections.

Thus the probability of winning is $\frac{120}{100000}=\frac{1}{833.333...}$. (Actually, I think this makes the odds of winning $1:832.333...$.)

paw88789
  • 40,402