Question

A game costs $3 to play. We are given a box with: 4 balls labeled "0", 3 balls labeled "1", and 2 balls labeled "3"

We select 2 balls at random at the same time and we win money equal tot he sum of their results.

With W a random variable which represents the total amount, we earn on one play of the game.

Find the density function of W

Answer 1

Possible vlaues		Sum
0	0	0	2 balls labelled 0
0	1	1	1 ball labelled 0 and other 1
0	2	2	1 ball labelled 0 and other 2
1	1	2	1 ball labelled 1 and other 1
1	2	3	1 ball labelled 1 and other 2
2	2	4	2 balls labelled 2
X	P(X)
0	0.166667
1	0.333333
2	0.222222
2	0.083333
3	0.166667
4	0.027778
	1

The excel formula used is

X	P(X)
0	=COMBIN(4,2)/COMBIN(9,2)
1	=4*3/COMBIN(9,2)
2	=4*2/COMBIN(9,2)
2	=COMBIN(3,2)/COMBIN(9,2)
3	=3*2/COMBIN(9,2)
4	=COMBIN(2,2)/COMBIN(9,2)
	=SUM(C13:C18)

Density function of W is

Amount W	P(W)
0	0.167
1	0.333
2	0.306
3	0.167
4	0.028
	1.000