Question

problem 9: expecting to lose (15) the knightrola lottery has been updated to allow choosing 6 numbers out of 63 on a $1 ticket. choosing 3 numbers correctly gives a $10 prize. choosing 4 numbers correctly gives a $50 prize. choosing 5 numbers correctly gives a $5,000 prize. if the lottery expects to precisely break even, how much is its prize for getting all 6 numbers correct?

Answer 1

The prize for getting all 6 numbers correct in the Knightrola Lottery is approximately $1,947,792.29.

To find the prize for getting all 6 numbers correct in the Knightrola Lottery, we can use the information provided in the question. The lottery allows choosing 6 numbers out of 63 on a $1 ticket, and choosing 3, 4, or 5 numbers correctly gives a $10, $50, or $5,000 prize, respectively. The lottery expects to precisely break even, which means that the total amount of money collected from ticket sales should be equal to the total amount of money paid out in prizes.

Let P be the prize for getting all 6 numbers correct. Then, we can set up an equation using the information provided and solve for the unknown prize amount. The equation for the expected value of the prize is:

`[ 1 * \binom{63}{6} = \left( \binom{6}{3} * 10 \right) + \left( \binom{6}{4} * 50 \right) + \left( \binom{6}{5} * 5000 \right) + P ]`

Simplifying the right-hand side of the equation, we get:

`[ 1 * \binom{63}{6} = 10 * 20 + 50 * 15 + 5000 * 6 + P ]`

`[ \binom{63}{6} = 20 + 750 + 30,000 + P ]`

`[ P = \binom{63}{6} - 30,770 ]`

Using a calculator, we can evaluate the expression on the right-hand side to get:

`[ P \approx 1,947,792.29 ]`

Therefore, the prize for getting all 6 numbers correct in the Knightrola Lottery is approximately $1,947,792.29.