Lottery Odds

To see the calculations of the odds and payouts, you can download my casino odds spreadsheet by right-clicking and selecting "Save target as".

To calculate the chances of winning the lottery, you need to calculate the number of possible combinations, then calculate the number of possible matches, and then divide the combinations by the matches.

Combinations of total numbers

The number of combinations is:


where:       x = number of balls used     
n = number of balls chosen

Example: 5 balls are picked out of 59.

combin(59,5) = 5,006,386

Combinations of matches

The number of ways to match is:

combin(n,m) * combin(x-n,n-m)

where:     x = number of balls used       
m = number of balls matched
n = number of balls picked    

Example: Match 4 out of 5 balls when 5 are picked out of 59.

combin(5,4) * combin(54,1) = 270

Calculating the Odds

To calculate the odds, you combine the two formulas above and by dividing the number of possible matches by the number of possible combinations. Then you divide that entire equation by 1 to find the chances.

1/((combin(5,4) * combin(54,1))/ combin(59,5)) = 1 in 18,542

Adjusting for a bonus ball

The bonus ball has a deceptive effect on lottery odds. This is because you have to pick a "specific" number to match the bonus ball. For a game with 39 bonus balls used, this makes it 39 times harder to win the jackpot. Because the bonus ball is separate from the regular balls, the calculation needs to be separate from the calculation of the regular balls. You need to make two adjustments whenever a lottery uses a bonus ball.

    Adjustment #1: Total Combinations

    In order to find out the new total number of combinations you need to multiply the previous equation by "combin(b,1)". Therefore, the new number of combinations is:


    where:    b = number of bonus balls used    

    Example: I'll use the Powerball lottery as an example. What are the odds of matching all 5 balls that are picked out of 59 - and matching the bonus ball when there are 35 bonus balls used.

    combin(59,5)*combin(35,1) = 175,223,510.00

    Adjustment #2: Number of matches

    The other adjustment only needs to be made if you are calculating the odds of hitting all the numbers except the bonus ball (i.e. you get the bonus ball wrong). To do this, you multiply the top half of the equation (the combinations of possible matches) by "combin(b-1,1)". This is all assuming that there is only 1 bonus ball to choose, which is the case.

    combin(n,m) * combin(x-n,n-m) * combin(b-1,1)

    Adding the two equations together we get:

    Bonus ball correct: (combin(n,m) * combin(x-n,n-m))/(combin(x,n)*combin(b,1) )
    Bonus ball wrong: (combin(n,m) * combin(x-n,n-m) * combin(b-1,1))/(combin(x,n)*combin(b,1) )

    Example: Again, using the Powerball as an example. What are the odds of matching all 5 balls that are picked out of 59 - and choosing 1 bonus ball out of 35 bonus balls.

    Bonus ball wrong:
    =1/((combin(5,5)*combin(54,0)*combin(34,1))/(combin(59,5)*combin(35,1))) = 1 in 5,138,133

    Bonus ball correct:
    =1/((combin(5,5)*combin(54,0))/(combin(59,5)*combin(35,1))) = 1 in 175,223,510.00

PowerBall Odds

Match Prize Odds Chances
5 balls + powerball Grand Prize 1 in 175,223,510.00 0.0000006%
5 balls $200,000 1 in 5,153,632.65 0.0000194%
4 balls + powerball $10,000 1 in 648,975.96 0.0001541%
4 balls $100 1 in 19,087.53 0.0052390%
3 balls + powerball $100 1 in 12,244.83 0.0081667%
3 balls $7 1 in 360.14 0.2776682%
2 balls + powerball $7 1 in 706.43 0.1415563%
1 ball   + powerball $4 1 in 110.81 0.9024217%
0 balls + powerball $3 1 in 55.41 1.8048434%
Odds of winning any prize   1 in 31.85 3.1400695%


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Lottery Odds

HPG ADMIN on February 27, 2013