MATH 115 – Sample Final Exam 2
1. What is the probability that three randomly-selected people were born on different
days of the week? (Assume that the chance of someone being born on a given day
of the week is 1/7).
(a) 1
343 (b) 1
27 (c) 30
343 (d) 30
49 (e) 35
81 (f) 1
3(g) 3
7(h) 1
720
2. What are the expected value µand the variance σ2of a random variable Xdis-
tributed on the interval [0,2] with the probability density function f(x)=x/2for
0≤x≤2?
(a) µ=4/3, σ2=2 (b)µ=4/3, σ2=2/9(c)µ=4/3, σ2=1/3
(d) µ=3/2, σ2=2 (e)µ=3/2, σ2=2/9
(f) µ=3/2, σ2=1/3(g)µ=1,σ2=2 (h)µ=1,σ2=1/3
3. Six different pairs of socks are put in the laundry (12 socks in all, and each sock
has only one mate), but only 7 socks come back. What is the expected number of
pairs of socks that come back? (Hint: This is the sum of the expectations of the
number, 0 or 1, of each individual pair that come back.)
(a) 7
22 (b) 15
8(c) 15
22 (d) 30
11 (e) 15
4(f) 21
11 (g) 23
8(h) 51
22
4. A student knows how to do 15 out of the 20 core problems for a given chapter. If
the TA chooses 3 of the core problems at random for a quiz, what is the probability
that the student knows how to do exactly 2 of them?
(a) 27
64 (b) 35
228 (c) 5
114 (d) 35
76 (e) 55
64 (f) 5
38 (g) 137
228 (h) 9
64
5. 6% of Type A spark plugs are defective, 4% of Type B spark plugs are defective,
and 2% of Type C spark plugs are defective. A spark plug is selected at random
from a batch of spark plugs containing 50 Type A plugs, 30 Type B plugs, and 20
Type C plugs. The selected plug is found to be defective. What is the probability
that the selected plug was of Type A?
(a) 3
4(b) 10
23 (c) 1
3(d) 2
3(e) 15
23 (f) 2
23 (g) 1
2(h) 7
10
6. The response time of communications network WEQ is an exponentially dis-
tributed random variable with mean response time of 0.1 seconds. A computer
has already been waiting for 0.2 seconds for a response from WEQ. What is the
probability that the computer will have to wait more than another 0.3 seconds, so
that the total response time is more than 0.5 seconds?
(a) 0.5 (b) 0.5 (c) 1
√5(d) e−10 (e) 0.3(f)e−2(g) e−3(h) e−5
