Question

In a contest run by a store, each customer whose purchase exceeds $100 is allowed to draw a discount coupon from a jar. At the beginning of the contest, the jar contains 30 slips for a 5% discount, 15 slips for an x% discount, and 5 slips for a 15% discount. If the expected value of the first draw from the jar is 6.6%, the value of x is __ . At one point in the contest, the jar contains 4 slips for a 5% discount, y slips for an x% discount, and 2 slips for a 15% discount. If the expected value on the next draw is 8%, the value of y is __ .

Answer 1

Answers:
x = 7%
y = 2 slips

Explanation:

The expected value is the result of the sum of each value times its probabilities:

Expeted value = probability 1 × value 1 + probability 2 × value 2 + probability 3 × value 3 + .....


Case 1: at the beginning of the contest:
total number of slips: 30 + 15 + 5 = 50

probability 1 = 30/50

value 1 = 5%


probability 2 = (15/50)
value 2 = x%


probability 3 = (5/50)

value 3 = 15%


⇒  Expected value = 6.6% = (30/50) 5% + (15/50)x% + (5/50)15%
⇒ (15/50)x% = 6.6% - (30/50)5% - (5/50)15%
⇒ (15/50) x% = 2.1% 
⇒ x% = (50 / 15) 2.1% = 7%

Answer: 7%

2) Case 2: at one point, ...


Yet, the equation for the expected value is:

Expeted value = probability 1 × value 1 + probability 2 × value 2 + probability 3 × value 3 + .....

Only the probabilities have changed, but the discounts are the same. This is x% is the same value found above: 7%.

The total number of slips now is 4 + y + 2 = 6 + y

And the expected value becomes:
8% = [ 4 / (6+y) ] 5% + [ y / (6 + y) ] 7% + [2 / (6 + y)] 15%

From which you obtain:

Mulitplying by 6+ y: 8% [6 + y] = 4×5% + y×7% + 2×15%

⇒ 8% y + 8%×6 = 4×5% + y 7% + 2×15%

⇒ 8% y - 7%y = 4×5% + 2×15% - 6×8%
⇒  0.01y = 0.2 + 0.3 - 0.48 = 0.02
⇒ y = 2