Question

Recall that the Powerball Lottery involves selecting five numbers from 1 to 69 for the white balls and then selecting one number from 1 to 26 for the red Powerball.

The Lottery does not give a prize for matching exactly two white balls without a matching red ball. Why do you think that prizes are not given for this combination? Justify your answer using probabilities.

Answer 1

White balls only-

Corresponding to white balls there are numbers from 1 to 69 i.e. in total 69 numbers.

Number of possible draws of 5 numbers from 69 numbers is

Number of possible draws in which exactly 2 numbers match

So, probability of matching exactly two numbers for white balls is given by

White balls and red ball-

There are 69 numbers for white balls and 26 numbers for red balls.

Probability of matching exactly two numbers for white balls and a number for red ball is given by

Comparison and explanation-

If the Lottery gives a prize for matching exactly two white balls without a matching red ball, then probability of giving a prize is 0.004262575 whereas if the Lottery gives a prize for matching exactly two white balls with matching of red ball, then probability of giving a prize is 0.0001639452 which is significantly lower than the previous.

Thus by just adding a constraint in the combination to win prize significantly reduces the probability of getting a prize by a player. In this manner the lottery arranging organisation can achieve higher profit. That is the reason to use such combination.