There are 4 balls, two red and the two blue. You choose a color each time and a random ball is picked from the sample (w/o replacement). If your choice and the random pick matches, you gain one dollar. Repeat this four times. How much would you pay to play this game
My Answer: We have a probability of 0.5 of getting the first guess right, and the second given we go for the one we didnt choose in the first trial is 2/3 and from there we would have 0.5 + 1 expected value, so my answer is $1(0.5) + 1(2/3) + 1(0.5) + 1 = 8/3$
Is this correct?
Best Answer
The optimum strategy is to pick the color with more balls or pick a random (you can also arbitrarily) color if they are the same number. Without loss of generality we can assume the first ball is red. So there are three outcomes with probability 1/3 each and expected winnings: \begin{align} \text{Outcome},&\quad\text{Winning}\\ RRBB, &\quad1/2+0+1+1 = 5/2\\ RBRB, &\quad1/2+1+1/2+1 = 3 \\ RBBR, &\quad1/2+1+1/2+1 = 3\\ \end{align} So the expected gain is (3+3+5/2)/3=17/6.