Practice questions for Test 1

1. Consider the sample data set
Practice questions for Test 1
51 42 38 10 27 65 51 34 39 17 35 43 13
33 57 21 37 63 45 22 49 31 43 64 47 22
(a) Construct a stem-and-leaf display of the data.
(b) Complete the following frequency table.
Class Boundaries
10 to < 20
20 to < 30
30 to < 40
40 to < 50
50 to < 60
60 to < 70
Frequency
TOTAL 26
(c) Construct a frequency histogram for the data set using the boundaries of part (b).
(d) Find the sample mean ¯x, median ˜x, variance s 2 and standard deviation s.
2. Let E and F be two events such that P (E) = 0.3 and P (F ) = 0.5. Compute P (E ∪ F ) if
(a) E and F are mutually exclusive.
(b) E and F are independent.
3. Let A and B be two events such that
(a) Compute P (A ∩ B).
(b) Compute P (B | A).
(c) Compute P (A | A ∪ B).
P (A) = 0.6, P (B) = 0.3, and P (A ∪ B) = 0.8.
4. Let A, B, and C be three independent events with
Compute P (A ∪ B ∪ C).
P (A) = 1/2, P (B) = 1/3, and P (C) = 2/3.
5. In how many distinct ways can a true-false test consisting of 9 questions be answered What
is the probability of getting exactly 6 questions right by guessing
6. In how many ways can 5 starting positions on a basketball team be filled from 8 players who
can play any positions
7. In a poker hand consisting of 5 cards, find the probability of holding
(a) 3 aces;
(b) 4 hearts and 1 club.

Math 264 Practice questions for Test 1 Page 2 of 2
8. A small town has one fire engine and one ambulance available for emergencies. The probability
that the fire engine is available when needed is 0.97 and the probability that the ambulance
is available when called is 0.94. In the event of an injury resulting from a burning building,
find the probability that both vehicles will be available. Clearly state what hypothesis you are
making.
9. A fire-detection device uses three temperature-sensitive cells acting independently of one another
in such a manner that any cell can activate the alarm. Each cell has a probability p = 0.85
of activating the alarm when the temperature reaches 100 ◦ C or more. Find the probability that
the alarm will function when the temperature reaches 100 ◦ C.
10. Assume that the probability of hitting a target is 1/5 and that 30 shots are fired independently.
(a) What is the probability of the target being hit exactly 8 times
(b) What is the probability of the target being hit at least twice
(c) What is the expected number of times that the target will be hit
11. The number of accidents occurring on a certain highway each day is a Poisson random variable
with λ = 3. Find the probability that there will be 3 or more accidents today.
12. Compare the Poisson approximation with the binomial probability for P (X = 6) if X is a
binomial with n = 500 and p = 0.01.
13. From a lot of 50 components, 6 are chosen at random for testing. If the lot contains 8 defective
components, what is the probability that
(a) all 6 components will be working.
(b) at most 2 of the 6 will be defective.
14. One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and
and 5 black balls. One ball is drawn from the first bag and placed in the second. What is the
probability that a ball now drawn from the second bag is black
15. A certain steroid detection test is 99.5% effective, meaning that it will show a positive result for
99.5% of all steroid users. The test also gives a “false positive” result for 2% of the non-user,
meaning that the probability of testing positive for a non-user is 2%.
A confidential survey of Olympic swimmers tells us that about 10% of them are using steroids.
Suppose one swimmer tests positive but claims he is not using steroids, what is the probability
that he is telling the truth
16. Consider a random variable X with the following probability distribution.
(a) Find the expected value of X.
x 1 2 3 4 5 6
p(x)
2
15
(b) Find the expected value of X 2 .
3
15
2
15
(c) Find the variance and standard deviation of X.
(d) Find the variance and expected value of the random variable Y = 5X + 3
3
15
4
15
1
15