Question

Suppose the owner of a local apple orchard has 4 half-bushels of apples to sell. He loads them on his truck, and drives his route, stopping at stores along the way attempting to sells the apples. Suppose:

- Each half-bushel is sold for $40.

- It will cost $15 of gas to drive the route.

Historically, the orchard owner knows he will have a 40% chance of making no sales, a 30% chance of selling 1 half-bushel, a 10% chance of selling 2 half-bushels, and a 10% chance of selling 3 half-bushels.

(a) Calculate the orchard owner’s expected profit from driving the route.

(b) Calculate the standard deviation of the profit from driving the route.

Answer 1

a) The orchard owner's expected profit from driving the route is $33.

b) The standard deviation of the profit is $27.28

Explanation:

a) Data and Calculations:

Apples to sell = 4 half-bushels

Price of each half-bushel sold = $40

Sales revenue = $160 ($40 *4)

Cost of driving the route = $15

Expected values:

Event Probability Revenue Profit Expected Profit

No Sales 40% $0 ($15) ($0 - $15) ($6.00)

Selling 1 half-bushel 30% $40 $25 ($40 - $15) $7.50

Selling 2 half-bushels 10% $80 $65 ($80 - $15) $6.50

Selling 3 half-bushels 10% $120 $105 ($120 - $15) $10.50

Selling 4 half-bushels 10% $160 $145 ($160 - $15) $14.50

Total expected profit = $33

Event Expected Profit Mean Squared

Difference

No Sales ($6.00) -39 1,521

Selling 1 half-bushel $7.50 -25.5 650.25

Selling 2 half-bushels $6.50 -26.5 702.25

Selling 3 half-bushels $10.50 -22.50 506.25

Selling 4 half-bushels $14.50 -18.50 342.25

Sum of squared differences 3,722

Mean of squared differences = 744.4 (3,722/5)

Standard deviation = square root of the mean

= 27.28

= $27.28