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Three different numbers are chosen at random from the set {1, 2, 3,..., 10}. Find the expected value of the largest number.

 Mar 16, 2020
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Each of the following possibilites has 6 different arrangements. We don't need to consider this to get the final answer.

 

If the largest number is 3, there is 1 possibility: (1,2,3).

If the largest number is 4, there are 3 possibilities: (1,2,4), (1,3,4), (2,3,4).

If the largest number is 5, there are 6 possibilities: (1,2,5), (1,3,5), (1,4,5), (2,3,5), (2,4,5), (3,4,5).

If the largest number is 6, there are 10 possibilities.

If the largest number is 7, there are 15 possibilities.

If the largest number is 8, there are 21 possibilities.

If the largest number is 9, there are 28 possibilities.

If the largest number is 10, thre are 36 possibilities.

 

To find the total number of possibilities, add: 1 + 3 + 6 + ... + 36  =  sum1

 

Now, we need to find the sum of the value times the number of possibilities:

    1 x 3  +  3 x 4  +  6 x 5  +  10 x 6  + ... +  36 x 10   =  sum2

 

To find the expected value, divide  sum2  by  sum1.

 Mar 17, 2020

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