Hey dumbledore.

In this problem you have a joint distribution where one variable is Win/Lose and the other is Good/Bad in terms of economic conditions. We assume that each variable is independent which gives the joint distribution P(X = Win/Lose, Y = Good/Bad) = P(X=Win/Lose)*P(Y=Good/Bad).

So for the expected profits, we need to take into account the total joint distribution which has four possibilities (unless specified otherwise, but your question has not).

So the expectation is E[E[X|Y]] = 840,000,000*(0.5)*(0.3) + 210,000,000*(0.7)*(0.5) + 150,000,000*(0.5)*(0.3) - 500,000,000*(0.3)*(0.5) = 147,000,000 based on the calculation in R:

> 840*0.3*0.5 + 210*0.7*0.5 + 150*0.3*0.5 - 500*0.3*0.5

[1] 147

Remember that you need to take into account all possibilities not just the situation where you win.

The situation where you win is basically E[Expected Winnings|You win the tribunal] which is what you were looking at but you can't assume you won unless specifically informed, and it says that there was no outcome yet and that the prediction was 50/50.

Other expectations include things like E[Expected Winnings|Bad economic conditions] as an example.