A lawyer sometimes represents client for a contingency fee T

A lawyer sometimes represents client for a contingency fee. The lawyer only gets paid for services rendered if the client wins the case. Suppose a client is suing someone for $400,000 and the lawyer\'s fee is 10% of the settlement. The lawyer will spend $2,000 preparing the case and believes there is a 20% chance of winning the case. Find the lawyer\'s expected profit.

Solution

The lawyer wins or loose, so there are only two outcomes of this:
His expected win is 20% and losing 80%
When he wins, he earnes 10% of $ 400.000 - $ 2.000
and when he loses his earning is - $ 2.000.
So the expected gain is 0.2 * 0.10 * $ 400.000 - 0.2 * $ 2.000 - 0.8 * $ 2.000
$ 8.000 - $ 2.000 = $ 6.000