Question

Here are the basic rules of Club Keno:

• You choose how many numbers you will pick. You can pick anywhere from 1 to 10 different numbers.
• Pick your numbers between 1 and 80.
• A drawing is held in which 20 numbers are picked.
• Depending on how many of your numbers come up in the drawing, you win various amounts of money.

In a previous assignment we saw the number of ways to choose exactly x correct from a total of n picks, you would do:

_nC_x * _80-nC_20-x

We also saw the total number of possible outcomes in Club Keno are ₈₀C_20.

Four Numbers - Expected Value

According to the Michigan LotteryOpens externally the best odds of winning are when you pick four numbers. If all four numbers come up then you win $72 for each dollar you bet. If three numbers come up then you win $5 for each dollar you bet. If two numbers come up then you win $1 for every dollar you bet (net winnings are zero). Otherwise, you lose the money you bet.

Fill out the following table, assuming a $1 bet. Enter probabilities as decimals, rounded

Outcome	Probability (Use Scientific Notation)	Net Value (Don't forget to account for the $1 bet)	Product (Round to 3 decimals)
4 correct		$	$
3 correct		$	$
2 correct		$	$
0 or 1 correct		$	$

What is the expected value if you choose four numbers? $

Answer 1

Outcome	Probability	Net Value	Product
4 correct	3.06E-03	$72	$0.221
3 correct	4.32E-02	$5	$0.216
2 correct	2.13E-01	$1	$0.213
0 or 1 correct	7.41E-01	-$1	-$0.741
Sum			-$0.092

Thus, expected value if we choose n = 4 number = -$0.092.

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