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 Lottery 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.

Outcome	Probability (Use as many decimal places as you can)	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? $

Kicker

Another option in Club Keno is to play the "Kicker." This is an option that doubles your bet, but you can get a random multiplier to your winnings. About 2/3 of the time there is no multiplier, though. Let's consider what the expected value is when playing the kicker while only betting on only one number.

Outcome	Kicker	Probability	Net Value
1 Correct	x1	0.15625	$0
1 Correct	x2	0.046875	$2
1 Correct	x3	0.0234375	$4
1 Correct	x4	0.01171875	$6
1 Correct	x5	0.00585938	$8
1 Correct	x10	0.00585938	$18
0 Correct	Any	0.75	-$2

What is the expected value when using the Kicker option and choosing one number? $ (Round to the nearest cent.)

If you had not played the kicker, the table would be:

Outcome	Probability	Net Value
1 Correct	0.25	$1
0 Correct	0.75	-$1

If you had not played the kicker, what would the expected value be?