Lottery Mathematics - Powerballs and Bonus Balls

Powerballs and Bonus Balls

Many lotteries have a powerball (or "bonus ball"). If the powerball is drawn from a pool of numbers different from the main lottery, then simply multiply the odds by the number of powerballs. For example, in the 6 from 49 lottery, if there were 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 × 10, or 566.6 (the probability would be divided by 10, to give an exact value of 8815/4994220). Another example of such a game is Mega Millions, albeit with different jackpot odds.

Where more than 1 powerball is drawn from a separate pool of balls to the main lottery (for example, in the Euromillions game), the odds of the different possible powerball matching scores should be calculated using the method shown in the "other scores" section above (in other words, treat the powerballs like a mini-lottery in their own right), and then multiplied by the odds of achieving the required main-lottery score.

If the powerball is drawn from the same pool of numbers as the main lottery, then, for a given target score, one must calculate the number of winning combinations, including the powerball. For games based on the Canadian lottery (such as the United Kingdom's lottery), after the 6 main balls are drawn, an extra ball is drawn from the same pool of balls, and this becomes the powerball (or "bonus ball"), and there is an extra prize for matching 5 balls and the bonus ball. As described in the "other scores" section above, the number of ways one can obtain a score of 5 from a single ticket is or 258. Since the number of remaining balls is 43, and the ticket has 1 unmatched number remaining, 1/43 of these 258 combinations will match the next ball drawn (the powerball). So, there are 258/43 = 6 ways of achieving it. Therefore, the odds of getting a score of 5 and the powerball are = 1 in 2,330,636.

Of the 258 combinations that match 5 of the main 6 balls, in 42/43 of them the remaining number will not match the powerball, giving odds of = 3/166,474 (approximately 55,491.33) for obtaining a score of 5 without matching the powerball.

Using the same principle, to calculate the odds of getting a score of 2 and the powerball, calculate the number of ways to get a score of 2 as = 1,851,150 then multiply this by the probability of one of the remaining four numbers matching the bonus ball, which is 4/43. Since 1,851,150 × (4/43) = 172,200, the probability of obtaining the score of 2 and the bonus ball is = 1025/83237. This gives approximate decimal odds of 81.2.

The general formula for matching balls in a choose lottery with one bonus ball from the pool of balls is:

The general formula for matching balls in a choose lottery with zero bonus ball from the pool of balls is:

The general formula for matching balls in a choose lottery with one bonus ball from a separate pool of balls is:

The general formula for matching balls in a choose lottery with no bonus ball from a separate pool of balls is: