Question

8. (5 marks) A game is to be played by selecting 5 balls from a lot of 20 containing 12 red balls and 8 black balls. Let W=3^x-17 be your win amount, where X represents the number of black balls drawn. Calculate the expected value and variance of W.

Answer 1

So, There are two categories of balls, red (12) and black (8). X denote the number of black balls drawn if 5 balls drawn at random out of 20 balls.

So , X ~ Hypergeometric ( 20 , 8 , 5)

x =0,1,2,.....5

20-8 (53)

Now ,

So ,

EW-E341-17

and

x	P[X=x]	3^X	3^X. P[X=x]		(3^x-17)^2	(3^x-17)^2. P[X=x]
0	0.051084	1	0.051084		256	13.07739938
1	0.255418	3	0.766254		196	50.0619195
2	0.397317	9	3.575851		64	25.42827657
3	0.23839	27	6.436533		100	23.83900929
4	0.05418	81	4.388545		4096	221.9195046
5	0.003612	243	0.877709		51076	184.4850361
		Total	16.09598		---	518.8111455

So, E[W] = 16.09-17 = -0.91  

V[W] = E[W²] - E²[W] = 518.81 - 16.09² = 259.92