Question

Jim decides to start a small nonprofit business of renting out motor scooters to help out his area residents. He puts in his own money and buys 5 motor scooters, each priced at $3,000. He incurs no other costs because he keeps the motor scooters in his own garage. The motor scooters last for 5 years. The number of motor scooters and the probability that they would be rented per month is shown in the table.
Number of Scooters 0         1          2           3            4       5
Probability              1/32     5/32     10/32     10/32     5/32    1/32


At what price per month should Jim rent out a scooter in order to break even?

A) 50
B)150
C)200
D)250
E)300

Answer 1

First, find the expected number of scooters rented per month:

As the data is symmetrical, E(X) (the expected value) is the middle value. So, on average, 2.5 scooters should be taken per month.

His total costs were 5 * 3000 = $15,000

So, to break even, he needs to make $15,000.

He will be selling for 5 years, or 60 months.

As a result, he needs to make 15000/60 = $250/month

As he is selling 2.5 scooters on average, he needs to rent each for:

$250/2.5 = $100/month

Answer 2

$the\ expected\ value:\ \ \ E[X] = x_1p_1 + x_2p_2 + \dotsb + x_kp_k \\\\ E[X] =0\cdot \frac{1}{32} +1\cdot \frac{5}{32} +2\cdot \frac{10}{32} +3\cdot \frac{10}{32}+4\cdot \frac{5}{32}+5\cdot \frac{1}{32}= \frac{80}{32} =2.5\\\\5\ motor\ scooters\ \ \rightarrow\ \ \ 5\cdot \ 3.000=\ 15.000\\\\5\ years=5\cdot 12\ months=60\ months\\\\2.5\ motors\ monthly\ \ \rightarrow\ \ 2.5\cdot60=150\ motors\ at\ 5\ years$

$x\ \ \rightarrow\ \ the\ price\ per\ month\ (to\ break\ even)\\\\150\cdot x=\ 15.000\ \ \ \Rightarrow\ \ \ x= \ 100\\\\Ans.\ \ \ 100$