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We have 7 cards labeled 1 to 7 and we draw 4 of them. Let X...

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.

Math   Statistics-And-Probability   
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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

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