What is the minimum expected amount in following? Suppose X borrowed $100 from you. Now, there is 0.8 probability that X will return >=50% of the loaned amount and 0.2 probability that <50%  of the amount.
Now, is it right to say that expected minimum amount is :
expected minimum amount= (0.8*(50% of $100) + 0.2*0)

which will be \$40 according to above formula, shouldn't it be just minimum expected amount is \$50 with 80% probability ?
I've serious doubt in this, how can the probability of getting minimum 50% of the amount(\$50) results in that minimum amount expected is $40 ?
 A: The question is unclear. Before anything, we should note that the actual minimum possible is 0 which may happen in the 20% branch.
There are two distributions at work here:

*

*Which branch you follow

*What is paid in each branch

The only way I understand the problem would be asking for the mean value assuming question 2 is always answered with a minimum payment. Once you assume stochasticity in question 2, it must be phrased as probability of getting less than X, since we KNOW what the minimum payment is for each branch: 0 and 50 respectively. In the case where we don't know the true lower end of a distribution from which we have samples is when we can start talking about the distribution of a minimum based on observations.
That being said, given the two branches, there is a 20% chance of getting a minimum of 0 and an 80% chance of getting a minimum of 50, so if one knows one will be receiving a minimal payout then the expected value of that payout is $40. That is not the expected payout over all possibilities---that depends on the distribution of payments in the above question 2. It is also not the minimum possible amount, that is clearly 0.
