Question

We have 7 cards labeled 1 to 7 and we draw 4 of them. Let X be the minimum number among our 4 cards. Find the expected value of X.

Answer 1

These are the following outcomes, where X = min(x1,x2,x3,x4)

x1	x2	x3	x4	X
1	2	3	4	1
1	2	3	5	1
1	2	3	6	1
1	2	3	7	1
1	2	4	5	1
1	2	4	6	1
1	2	4	7	1
1	2	5	6	1
1	2	5	7	1
1	2	6	7	1
1	3	4	5	1
1	3	4	6	1
1	3	4	7	1
1	3	5	6	1
1	3	5	7	1
1	3	6	7	1
1	4	5	6	1
1	4	5	7	1
1	4	6	7	1
1	5	6	7	1
2	3	4	5	2
2	3	4	6	2
2	3	4	7	2
2	3	5	6	2
2	3	5	7	2
2	3	6	7	2
2	4	5	6	2
2	4	5	7	2
2	4	6	7	2
2	5	6	7	2
3	4	5	6	3
3	4	5	7	3
3	4	6	7	3
3	5	6	7	3
4	5	6	7	4

PMF of X is:

Hence, expected value of X is:

1*20/35 + 2*10/35 + 3*4/35 + 4*1/35 = 56/35