Question

Three of six numbers 1, 2, . . . , 6 are randomly erased. Let X be the median of three remaining numbers. Find EX.

Answer 1

There are a total of 6c3 = 20 ways in which 3 numbers can be erased from the 6 numbers. The median in each case is computed here as:

Numbers Left	X
123	2
124	2
125	2
126	2
134	3
135	3
136	3
145	4
146	4
156	5
234	3
235	3
236	3
245	4
246	4
256	5
345	4
346	4
356	5
456	5

Now each of the 20 cases above are equally likely. Therefore the expected median is computed here as:

E(X) = (2 + 2 + 2 + .... + 5 + 5) / 20 = 70/20 = 3.5

Therefore 3.5 is the required expected median value here.

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