Expectation of lost ball number

Question

I have expectation problem that sounds like this:

In the box we have 5 balls numbered U={2,2,2,3,3}. We lost one ball and we don't know which one. Then one ball was taken out from the box which number is X. Calculate E[X]

I've tried to solve it like this:

H2 - '2' ball has been lost; H3 - '3' ball has been lost.

Then I calculated probabilities:

A - '2' ball was taken B - '3' ball was taken.

P(A) = P(H2)*P(A|H2)+P(H3)*P(A|H3)= 15/20

P(B) = P(H2)*P(B|H2)+P(H3)*P(B|H3)= 10/20

And then E[X]=(2*15/20)+(3*10/20)=60/20

However, I found somewhere that the answer might be 61/20. But I'm not sure.

So I would really like to know if my solve is correct. Thank you!

Answer 1

You have the correct formula, but miscalculated.

$P(A)=P(A\mid H_2)~P(H_2)+P(A\mid H_3)~P(H_3)\\\qquad= \tfrac 24\cdot\tfrac 35+\tfrac 34\cdot\tfrac 25\\\qquad =\tfrac 35$

Similarly $P(B)=\tfrac 25$

Reevaluate your expectation with these values.

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Also, think about what this is telling you about the effect of the missing ball on the expectation for the second ball.

Answer 2

The "lost ball" is a distractor, completely irrelevant. If two balls are drawn from the box, you need to find the expected value of the second...

...and then "second" is just a distractor too. You need to find the expected value if one ball is drawn from the box. Can you do this?