Butch2030 wrote:
The odds on the back of a play slip might not the be the true odds of the game.
***Lets look at the back of a play slip for the IL Power Ball Game.
***Match 5 of 6 + PB = 1 : 292,201,338
***Match 5 of 6 = 1 : 11,688,054 (note: 11,688,054 x 26 PB = 303,889,404)
***Match 4 of 6 = 1 : 36,525
***Match 3 of 6 = 1 : 580
***
***Here are your true odds:
***Match 5 of 6 + PB = 1 : 292,201,338
***Match 5 of 6 = 1 : 11,238,513 (note: 11,238,513 x 26 PB = 292,201,338)
***Match 4 of 6 = 1 : 864,501
***Match 3 of 6 = 1 : 52,394
The IL odds are correct.
Of course, no one who matches 5 of 5 and the PB would claim the match 5 of 5 prize instead.
So there are 25 of 292,201,338 combinations that match 5 of 5 and not the PB. So the odds are 1 in 292,201,338 / 25 = 1 : 11,688,053.52. That is indeed how it appears on the powerball.com website. Apparently IL chooses to round to an integer (11,688,054).
Similarly, there are COMBIN(5,4)*64*25 = 8000 of 292,201,338 combinations that match 4 of 5 and not 5 of 5 and the PB. So the odds are 1 in 292,201,338 / 8000 = 1 : 36,525.16725. That rounds to 36,525.17 (powerball.com) or 36,525 (IL).
To explain: there are COMBIN(5,4) = 5 ways for our 5 numbers to match 4 and not 5 of 5 numbers. That leaves 69 - 5 = 64 choices for the 5th lottery number so that our 5 numbers match 4 and not 5 of any lottery selection. And again, there are 25 ways to not match the PB.
Similarly, there are COMBIN(5,3)*COMBIN(64,2)*25 = 504,000 of 292,201,338 combinations that match 3 of 5 and not 4 of 5 and not 5 of 5 and the PB. So the odds are 1 in 292,201,338 / 504,000 = 1 : 579.76455952381 (approx), which rounds to 579.76 (powerball.com) or 580 (IL).