Consider the following game: Three cards are labeled $1, $4, and $7. A player pays a...

Consider the following game: Three cards are labeled $1, $4, and $7. A player pays a $9 entry fee, selects 2 cards at random without replacement, and then receives the sum of the winnings indicated on the 2 cards.

a) Calculate the expected value and standard deviation of the random variable "net winnings" (that is, winnings minus a $9 entry fee)

b) Suppose a 4th card, labelled k, is added to the game but the player still selects two cards without replacement. What is the value of k which makes the game fair (i.e makes expected net winnings = $0)

Expert Solution

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a).

P (X1)	1/3
P (X2)	1/2

X1	X2	Y (X1 + X2)
1	4	5
1	7	8
4	1	5
7	1	8
4	7	11
7	4	11

therefore P (Y) = P (X1) * P (X2)

Y	Freq.	P (Y)	W (Y - 9)	P (W)	W * P (W)	W ^2	W ^2 * P (W)
5	2	1/3	-4	1/3	-1 1/3	16	5 1/3
8	2	1/3	-1	1/3	- 1/3	1	1/3
11	2	1/3	2	1/3	2/3	4	1 1/3
	sum	1		1	-1		7

b ).

X1'	X2'	Y' (X1' +X2')
1	4	5
1	7	8
1	K	K + 1
4	K	K + 4
4	1	5
4	7	11
7	4	11
7	K	K + 7
7	1	8
K	1	K + 1
K	7	K + 7
K	4	K + 4
		72 + 6K

P (X1')	1/4
P (X2')	1/3

Y'	Freq.	P (Y')	W (Y'- 9)
11	2	1/6	2
5	2	1/6	-4
8	2	1/6	-1
K + 1	2	1/6	K - 8
K + 4	2	1/6	K - 5
K + 7	2	1/6	K - 2
Sum	12	1

milcah answered 2 hours ago

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