What Are the Odds of Winning the Lottery

June 14, 2009

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We take you back to math class in today’s edition of The Soap Boxers, as we discuss probability.

First of all, if someone tells you that you have a million to one odds of winning a lottery, they are wrong.  Your odds are actually a million to one AGAINST you winning.  Another way to phrase this is that you have a one IN 1,000,001 chance of winning.  But we’re being too picky, so let’s get down to the nitty gritty.

Let’s examine the multi-state Powerball game.  Powerball chooses 5 winning white balls from a group of 59 white balls, and then one red ball (Powerball) from a separate set of 39 red balls.  To win the jackpot, you must match all 5 white balls (in any order) plus the Powerball.

To eliminate confusions, I will refer to the numbers on your tickets as “numbers” and the winning numbered balls as “ball”.  Let’s use an example winning combination of 04, 18, 26, 31, 50 with a Powerball of 17.

  • First number: The first number on your ticket has a 5 in 59 chance of matching one of the winning white balls.  There are 5 winning white balls and 59 total balls.  It isn’t necessary to match the white balls in order, so matching any winning white balls is sufficient.  Let’s say that the first number on the ticket is 26.
  • Second number: The odds of a match drops to 4 in 58.  Why the change in odds?  Because the first number on the ticket was 26 – the second number can’t duplicate this!  We reduce the number of possible winning numbers to 4 possibilities (they are 04, 18, 31, 50) and the total number of balls remaining to 58 – because there are actually only 58 balls remaining in the selection pool.  The basic lesson is that the selection of the white balls are not independent events – the selection of each balls affects the pool of balls that are used for subsequent selections.
  • Third number: The odds are 3 in 57, as there are two fewer balls in the hopper than when we began, and two fewer winning possibilities.
  • Fourth number: Odds are 2 in 56
  • Fifth number: Odds are 1 in 55
  • Powerball:  The Powerball is completely independent from the other balls.  The odds are 1 in 39.

OK, we have the odds; what do we do with them?  We multiply them:

5/59 * 4/58 * 3/57 * 2/56 * 1/55 *  1/39

or

120/23,429,886,480

or

1/ 195,249,054

If you want to get really fancy, crack out factorials:

5!/((59! / 54!) * 39)

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