I'm a little confused about the following lecture slide, which is written as follows.
A risk-neutral individual chooses among pairs of gambles
{X1,X2} such that Xi = 10 with probability 1.
{Y1, Y2} such that Yi = 0 or 16 with probability ½ each.
{X1, Y1}
The expected values of the higher draws are:
E[max{X1,X2}] = 10
E[max{Y1,Y2}] = ¼(0) + ¾(16) = 12
E[max{X1,Y1}] = ½(10) + ½(16) = 13
The expected values of the lower draws are:
E[min{X1,X2}] = 10
E[min{Y1,Y2}] = ¾(0) + ¼(16) = 4
E[min{X1,Y1}] = ½(0) + ½(10) = 5
Specifically, I'm confused as to how to evaluate the expected values of maximum and minimum functions, where the "1/4" and "3/4" weights came from in the E[max/min(Y1,Y2)], and where the "1/2" weights came from in E[max/min(X1,Y1)]. The prof. gave the example of choosing between players with different values for a sports draft, so any clarification as to what's going on here in those terms would also be helpful.
Thanks!
consider $E[\rm{max\ \{y_1,y_2\}}]$. Prob $y_1$=0 is 1/2, probability $y_2$=0 is 1/2, so the probability they are both 0 is 1/4. If that case happens, then the max is 0. Probability that something else happens is 1-1/4=3/4, and something else is either $y_1$ is 16 and $y_2$ is 0, or vice versa, or both $y_1$ and $y_2$ are 16. In all those cases, the max is 16, so the expected value is 1/4 (0) + 3/4 (16) = 12. And the other problems follow a similar logic.