There are two blue balls and two red balls in a box. At each turn, you will guess the color of the ball you are about to randomly select. If you guess the color correctly, you receive a dollar. You continue to draw balls without replacement, guessing the color at each turn, until there are no balls left. What is the expected value of this game if you play optimally?

Answer: now, the expected value will be x = p1*0$ + p2*1$

where p1 is te probabilty of a fail and p2 the probability of succes.

Ok, we will have 4 steps here.

1) there are 4 balls, and if we chose a spesific colour, there are 50% chance of succes. x = 0.5$

2) there are 3 balls, but yo know that if in the first step you graved a blue ball, then here you have a 66% of getting a red one, so if you play optimaly, you will guess red. x = 0.6$

3a) now there are two posibilities, in the last step yo get the other blue ball, so now are two red balls in the box, and you have guaranted 2 bucks. x = 2$ (but you failed in the last step)

3b) if in 2 you get a red ball, then again you have a 50/50 chance for each colour. x = 0.5$

4) there's only one ball in the box, you get a dollar x = 1 $

so if you go with the 3b path, te expected value will be 2.6$

with 3a) x = 2.5$