A bank runs a contest to encourage new customers to open accounts. In the contest, each contestant draws a slip representing a different reward—$5, $3, or $x—from a jar. At the beginning of the contest the jar contains 60 slips for $5, 40 slips for $3, and 50 slips for $x.If the expected value of the first draw from the jar is $5.8, the value of x is__At one point in the contest, the jar contains 3 slips for $5, 7 slips for $3, and y slips for $x. If the expected value on the next draw is $6, the value of y is __
Accepted Solution
A:
Part 2:
We'll use x = 9 from part 1.
Again let, A = event that the $5 reward is drawn B = event that the $3 reward is drawn C = event that the $x reward is drawn (x is some positive number)
We can update event C to say C = event that the $9 reward is drawn
The probabilities change to P(A) = 3/(10+y) P(B) = 7/(10+y) P(C) = y/(10+y) where y is some positive whole number. It represents the number of slips in jar C
The net values are V(A) = 5 V(B) = 3 V(C) = x = 9
Like before, multiply the probabilities and net values to get P(A)*V(A) = (3/(10+y))*5 = 15/(10+y) P(B)*V(B) = (7/(10+y))*3 = 21/(10+y) P(C)*V(C) = (y/(10+y))*9 = (9y)/(10+y)
The results add up to [ 15/(10+y) ] + [ 21/(10+y) ] + [ (9y)/(10+y) ] (15+21+9y)/(10+y) (36+9y)/(10+y)
That last expression is the expected value. The expected value is also given to be 6, so set the two expressions equal to each other and solve for y. (36+9y)/(10+y) = 6 36+9y = 6(10+y) 36+9y = 6(10)+6(y) 36+9y = 60+6y 9y-6y = 60-36 3y = 24 3y/3 = 24/3 y = 8
The statement "y slips of $x" turns into "8 slips of $9" since x = 9 and y = 8.