1. LaTasha has started her own custom T-Shirt company, T\'s T\'s, which produces silk screened
T-shirts for various occasions and organizations; as well as a brisk walk-in trade. Her shop is
located in a small beach community in Florida. Since she has just started she is renting the
equipment from a large firm in Jacksonville. The cost of renting the equipment is $500. The
material used in one custom T-shirt costs $4.50, and LaTasha is selling the T-shirts for $12 each.
a. If LaTasha sells 50 shirts,
1. What will her total revenue be?
2. What will her total cost be?
b. How many shirts must LaTasha sell to break even?
c. What is the total revenue at the break-even point?
Solution
1. The total revenue is equal to the amount of income gained by selling the 50 shirts, not
accounting for any expenses (which net profit or net income values take into account). The T-
shirts are sold for $12 each, so one T-shirt costs $12, two T-shirts cost 2 times $12 = $24, and so
on. If 50 T-shirts are sold:
Total Revenue = (50 T-shirts) ($12 per T-shirt) = $600
2. Her total cost is equal to the amount she has to pay. First, she must rent equipment for $500,
which she pays, and she must also pay $4.50 for each T-shirt that she buys (and may or may not
sell). Her total cost is the sum of the costs of the equipment and the T-shirts:
Total Cost = $500 + (50 T-shirts) ($4.50 per T-shirt) = $500 + $225 = $725
b) In order to break even, the total revenue must equal the total cost. If x is the number of T-
shirts needed for LaTasha to break even, her total revenue must equal her total cost. Her total
revenue is given by 12x ($12 per each T-shirt), and her total cost is given by 500 + 4.50x ($500
rent plus $4.50 per each T-shirt). We want to solve for x when the total revenue and total cost are
equal:
Total Revenue = Total Cost
12x = 500 + 4.50x
7.50x = 500
x = 500/7.50 = 66.66666... 67 T-shirts
c) After breaking even, her total revenue, given by 12x, is found by multiplying 12 by the value
of x we just calculated: x 67
Total Revenue = 12x = 12 (67) = $804
(Note that, because we can\'t sell 0.666... of a T-shirt, x is rounded to 67 T-shirts. However,
when x = 67, the total revenue actually exceeds the total cost by a bit, so it isn\'t exactly
\"breaking even\", but its as close as possible that we can get while still selling an integer number
of T-shirts.).
1. Let D be the random variable with value equal to the number showi.pdf
1. 1. Let D be the random variable with value equal to the number showing on a toss of a
six sided die. Let L be the random variable with value equal to the least number of four
tosses of the die. Describe the probability functions D and L. That is, describe p (D
Solution
When I look at this problem, I see AE between D and r and likewise between L and r and O and
r. If this is some other sign, you can easily correct these answer.
1. P(D = r) = 1/6 for r = 1, 2, 3, 4, 5, 6
L is the minimum of the four tosses.
P(min of 4 tosses) = r = for r = 6, (1/6)^4 = 1/1296
For r <6, it is P(all tosses >= r) - P(all tosses >= r+1) = ((7-r)/6)^4 - ((7-(r+1))/6)^4 = ((7-r)/6)^4
- ((6-r)/6)^4
Note that this formula also works for r=6, so we may simply write
P(L=r) = ((7-r)/6)^4 - ((6-r)/6)^4 for r = 1, 2, 3, 4, 5, 6
We may give the values explicitly
P(1) = 1296/1296 - 625/1296 = 671/1296
P(2) = 625/1296 - 256/1296 = 369/1296 = 41/144
P(3) = 256/1296 - 81/1296 = 175/1296
P(4) = 81/1296 - 16/1296 = 65/1296
P(5) = 16/1296 - 1/1296 = 15/1296 = 5/432
P(6) = 1/1296 - 0/1296 = 1/1296
2.
There are many ways to solve this problem.
One ways is to list all 2^5 = 32 possibilities and count the number with each value of r.
2. One way, which is a bit more work, is the following.
Clearly, P(O = 0) = P(O = 5) = 1/2^5 = 1/32 (all 0s and all 1s implies 1 way for each).
P(O = 4) = 2/32 = 1/16 (either the first four or last four are all 1s).
P(O=1) = (there are 5 ways of only having 1 1; there are C(5,2) = 10 ways of having 2 1's, but
we remove the 4 with consecutive 1s is 10 - 4 = 6. There is one way to have three 1s not
consecutive, 10101. Thus, the total is 5 + 10 - 4 + 1 = 12
P(O = 1) = 12/32 = 3/8
P(O = 3) either the 3 middle numbers are all 1s, or the first 3 are 1s, the fourth is 0, and the fifth
is easier 0 or 1, or the last three are 1s, the second is 0, and the first is either 0 or 1. 1 + 2 + 2 = 5
P(O = 3) = 5/32
P(O=2) The easiest way to figure this is to simply substract the sum of the other probabilities
from 1. We get 1 - 1/32 - 1/32 - 1/16 - 3/8 - 5/32 = 11/32
We can calculate this a different way, thus checking the other answers.
We have the first two as 1s, the third as a 0, and the fourth and fifth anything (4 ways) + the
fourth and fifth 1s, the third 0, and the first two anything (4 ways) - the first and last two all 1s
and the third 0. Then, we have the first 0, the second and third 1s, the fourth 0, and the fifth 0 or
1 (2 ways), and the third and fourth 1s, the fifth and second 0s, and the first 0 or 1 (2 ways), for a
total of 4+4-1+2+2 = 11
Thus, P(O = 2) = 11/32; this matches our other calculation of it, which gives us faith in our other
answers.
P(O = 0) = 1/32
P(O = 1) = 3/8
P(O = 2) = 11/32
P(O = 3) = 5/32
P(O = 4) = 1/16
P(O = 5) = 1/2^5 = 1/32
