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%?
Best Answer
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 $1:832.333...$.)