Download Midterm 2 solutions

San José State University
Math 70, Fall 2004
Midterm 2 Solutions
Score
1
25
2
25
3
25
4
25
Total 100
1. (25 points) An urn contains two $10 bills, one $50 bill, and one $100 bill. A player draws
bills one at a time without replacement from the urn until a $100 bill is drawn. Then
the game stops. All bills are kept by the player.
(a) What is the probability of winning $120?
(b) What is the probability of winning all the bills in the urn?
Solution: The probability tree is:
1/2
$50
$100
$10
1/3
$10
1/2
START
1/3
1/3
1/4
2/3
1/2
1/2
1/4
$10
$100
$50
$100
1/2
$100
1/2
$10
1/2
$100
$10
$50
1/3
$100
$100
$100
$100
(a) A player wins $120 only if (s)he draws $10, followed by $10, followed by $100. The
probability of that event (corresponding to the top branch of the tree) is
P ($101, $102, $1003 ) =
111
1
= .
232
12
(b) A player wins all the bills if (s)he draws $100 in the fourth draw, which corresponds
to the three longest branches of the probability tree. Therefore,
P (winning all bills) = P ($101, $102, $503 , $1004) + P ($101, $502 , $103, $1004)
+ P ($501, $102, $103 , $1004)
111 111 121
=
+
+
232 232 432
1
.
=
4
2. (25 points) A game has an expected value to you of −$1. It costs $2 to play, but if
you win, you receive $20 (including your $2 bet), for a net gain of $18. What is the
probability of winning?
Solution: Let p be the probability of winning. Then the payoff table is
xi $18 −$2
pi
p 1−p
Therefore, the expected value of the game is
E(X) = 18p − 2(1 − p) = 20p − 2.
Since we are given that E(X) = −1, it follows that 20p − 2 = −1, so
p=
1
.
20
3. (25 points) A computer store sells three types of computers: 50% of them are brand P ,
25% are brand Q, and the rest are brand R. The store has found that 10% of the brand
P computers, 20% of the brand Q computers, and 5% of the brand R computers are
returned for service during the warranty period. If a computer is returned for service
during the warranty period, what is the probability that it is a brand P computer?
Solution: Let S denote the event that a computer is returned for service during the warranty
period. Then the probability tree is:
1/10
S
P
S0
1/2
1/4
START
1/5
S
Q
S0
1/4
1/20
S
R
S0
According to Bayes’s formula, the probability that a computer is brand P given that it
has been returned for service is:
P (P |S) =
1 1
2 10
1
1 1
11
+
+ 14 20
2 10
45
4
= .
9
4. (25 points) Animals in an experiment are to be kept under a strict diet. Each animal is to
receive, among other nutrients, 14 grams of protein and 6 grams of fat. The laboratory
technician is able to purchase two food mixes of the following compositions: mix A has
20% of protein and 8% of fat; mix B has 10% of protein and 5% of fat. How many
grams of each mix should be used to obtain the right diet for a single animal?
Solution: Let A, B be the required amounts (in grams) of mixes A, B, respectively. Then a
single animal will be getting 0.2A + 0.1B grams of protein and 0.08A + 0.05B grams of
fat. Therefore, we need to solve the following system of equations:
0.2A + 0.1B = 14
0.08A + 0.05B = 6.
Multiplying the first equation by 10 and the second by 100, we obtain
2A + B = 140
8A + 5B = 600.
Multiplying the first equation by −4 and adding it to the second equation, we get
2A + B = 140
B = 40.
Substituting B = 40 into the first equation yields A = 50. Therefore,
A = 50, B = 40.