My question is about winning a prize in the Powerball® lottery by matching only one of six numbers; i.e., the red Powerball® number.
Powerball® costs $2 per Play.
A play consists of selecting five numbers from 1 to 69 (with replacement) and one number from 1 to 26.
The winning numbers are determined by drawing one white ball from each of the five bins of white balls numbered 1 to 69 and one red ball, i.e., the Powerball®, from a bin of red balls numbered 1 to 26.
According to the Powerball® website (www.powerball.com), there are nine ways to win. The table below copied from their website describes the match requirements for each of the nine ways to win along with its corresponding prize and odds of winning.
The ninth way to win a prize is to match the number on the red Powerball® while not matching any of the numbers on the five white balls.
Can someone explain why the odds of winning by matching only the red Powerball® is not 1 in 26?
Thanks!
gfr92y
Intuitively, the probability of matching only the powerball is not $1/26$ because it's possible to match white balls at the same time.
We can add up the probabilities of all the prize combinations that include the powerball:
$$\frac{1}{38.32}+\frac{1}{91.98}+\frac{1}{701.33}+\frac{1}{14494.11}+\frac{1}{913129.18}+\frac{1}{292201338.00}=\frac{1}{26}$$
There are rounding errors in the above, but if the probabilities were given precisely then the summation would be exact.
(Side note: the figures given in the Powerball table are referred to as odds. Strictly, they are probabilities rather than odds. Example: the probability of rolling a $6$ on a fair $6$-sided die is $1/6$; the odds of the same event are $5:1$ against.)