Justify your answer and why you use each formula. DO NOT SKIP STEPS, EVEN SIMPLE

Question

Justify your answer and why you use each formula.

DO NOT SKIP STEPS, EVEN SIMPLE STEPS.

A lottery allows a player to choose 5 values out of 40. The goal of the lottery is to break even exactly, making its cash prizes equal to the amount of money spent by the contestants. Each ticket costs $1 to buy and players receive winnings if they match 3, 4 or all 5 numbers. If the payout for matching all 5 numbers is S250,000 and the payout for matching 4 numbers is $1000, how much should the payout for matching exactly 3 numbers be: to the nearest penny?

Explanation / Answer

The probability of matching all 5 numbers is

1/C(5,40) = 5! 35! / 40!

= 120/(36*37*38*39*40)

= 1/(12*37*38*39) = 1.51973*10^(-6)

The probability of matching exactly 4 numbers is

5*(5/40)*(4/39)*(3/38)*(2/37)*(35/36)

= 175/(13*38*37*36) = 2.65954*10^(-4)

Since the ratio of these two probabilities is 175 and not 250, the question seems to be irrational.

How are we to know whether the "big" winner has been overpaid, or whether the "almost" winners have been underpaid?

As we don't know how many people entered the lottery.

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