Math 166 Week-in-Review - © S. Nite 10/6/2012
WIR #5
Page 1 of 5
Week in Review #5 (2.3-2.4, 3.1)
In general, the probability of an event is P( E ) =
n( E )
.
n( S )
Distinguishable Permutations
Given a set of n objects in which n1 objects are alike and of one kind, n2 objects are alike
and of another kind,…, and, finally, nr objects are alike and of yet another kind so that
n1 + n2 + ... + nr = n then the number of permutations of these n objects taken n at a time is
given by
n!
n1!n 2 !...n r !
Bernoulli trials has two outcomes that are independent of each other. The two outcomes
are designated as “success” or “failure”; the probability of success is designated p and the
probability of failure q. (Note that q = 1 – p).
To find the probability of exactly k successes in n Bernoulli trials, use the product rule for
any one of the possibilities: pkqn-k. But there are C(n, k). So, the probability of obtaining
exactly k successive in n trials is
C(n, k) pkqn-k
Calculator Functions for binomial experiments (Bernoulli trials):
binompdf (#trials, p, #successes desired)
binomcdf (#trials, p, “at most” #successes)
Random Variables
A random variable is a rule that assigns a number to each outcome of a chance
experiment. It is a function with domain given by the set of outcomes of a chance
experiment and range contained in the set of real numbers.
A random variable is called finite discrete if it assumes only finitely many values.
A random variable is said to be infinite discrete if it takes on infinitely many values,
which may be arranged in a sequence.
A random variable is called continuous if the values it may assume comprise an interval
of real numbers.
Probability Distributions of Random Variables
The probability distribution of a random variable may be given in the form of a
formula or displayed in a table that gives the distinct (numerical) values of the random
variable and the probabilities associated with these values.
Histograms
A probability distribution of a random variable may be exhibited graphically by means of
a histogram.
In a histogram, the area of a rectangle associated with a value of a random variable X
gives precisely the probability associated with the value of X. This follows because each
such rectangle, by construction, has width 1 and height corresponding to the probability
associated with the value of the random variable. Another consequence arising from the
method of construction of a histogram is that the probability associated with more than
one value of the random variable X is given by the sum of the areas of the rectangles
associated with those values of X.
Math 166 Week-in-Review - © S. Nite 10/6/2012
WIR #5
Page 2 of 5
1. A parts code consists of 2 digits followed by 3 letters. If a code is selected at
random, what is the probability that no letters are repeated all the digits are odd?
(Round to 4 decimal places.)
2. An urn has 6 white and 3 red balls. A sample of 4 is selected randomly without
replacement.
a. What is the probability that exactly 4 are white and exactly 2 are red?
b. What is the probability that at least 3 of the balls are white?
3. A 3-card hand is drawn from a standard deck of 52 cards. Find the probability
that the hand contains exactly 2 of the same suit?
4. A group of 7 Aggies as well as 15 other people apply for a job. The company
will select 6 people to interview. What is the probability that exactly 5 Aggies will
be selected to interview? (Give your answer to 4 decimal places.)
5. An exam consists of 15 true/false questions and 10 multiple-choice questions
with 4 answers each. If a student guesses at every answer, what is the probability
that he or she will answer exactly 10 true/false and 8 multiple-choice questions
correctly? (Round your answer to three decimal places.)
6. An exam has 18 questions. The instructions on the exam say that you have to
answer exactly 12 questions. Questions are randomly selected. What is the
probability that a student answered exactly 7 of the first 12 questions? (Round your
answer to 4 decimal places.)
7. A shelf in an electronic store contains 60 flash drives. Seven of the drives are
defective.
a) If a customer selects 4 flash drives at random from the shelf, what is the
probability that they are all defective? (Round your answer to 5 decimal places.)
b) If a customer selects 4 flash drives at random from the shelf, what is the
probability that at least 2 are defective? (Round your answer to 4 decimal places.)
8. A fair die is rolled five times. What is the probability of obtaining exactly three
6’s? Round your answer to 4 decimal places.
Math 166 Week-in-Review - © S. Nite 10/6/2012
WIR #5
Page 3 of 5
9. If the probability that a certain basketball player will make a free throw is 2/7,
what is the probability that he will make exactly 5 out of 9free throws? (Assume
that the free throws are independent.) Round your answer to 4 decimal places.
10. Find the probability of exactly 4 failures in a Bernoulli experiment that is
repeated 12 times and p = 0.65. (Round your answer to 5 decimal places).
11. Suppose 16% of restaurants in a certain town have “slime in the ice machine.”
A health inspector randomly selects 5 of the restaurants for inspection. Round your
answers to 4 decimal places.
a) What is the probability that none of the restaurants have “slime in the ice
machine?”
b) What is the probability that one of the restaurants has “slime in the ice
machine?”
3) What is the probability that at least three of the restaurants have “slime in the ice
machine?”
12. A company finds that one out of five workers it hires turns out to be
unsatisfactory. If the the satisfactory performance of any hired worker is
independent of that of any other hired workers, and the company hires 36 people,
what is the probability that least 10 workers and fewer than 18 workers will turn
out to be satisfactory? Round your answer to 4 decimal places.
13. Thirty-five percent of the people browsing a certain store will make a purchase.
What is the probability that among twelve people who are browsing the store, at
least seven will make a purchase? Round your answer to 4 decimal places.
14. Four couples are attending an Aggie basketball game. They have tickets next to
each other all in the same row. If the tickets are randomly given to the 8 people,
what is the probability of each couple sitting together?
15. A student must answer 15 of the 18 questions on the exam. What is the
probability that a given students answers at most 7 of the first 10 questions?
16. Two-year-old Mick is playing with 7 blocks, with the letter G on 2 blocks, E on
one block, O on one block, R on one block, I on one block, and A on one block. If
he lines up the blocks in a row, what is the probability that he spells the word
GEORGIA?
Math 166 Week-in-Review - © S. Nite 10/6/2012
WIR #5
Page 4 of 5
17. Eighteen students are all applying for 4 different scholarships. What is the
probability that one particular student will get at least 5 scholarships?
18. A committee of 8 students is to be chosen from a group of 12 freshmen, 10
sophomores, and 9 juniors. Find the probability that
a) The committee has all juniors.
b) The committee has a majority of freshmen.
c) Exactly 2 of 3 friends are on the committee.
19. A random variable X is the number of times a student takes the TAKS test
before passing.
a) Give the range of values that X may assume.
b) Classify the random variable (finite discrete, infinite discrete, or continuous).
20. A random variable X is the number of cards drawn from a well-shuffled desk of
52 cards until a King is drawn.
a) What values may X assume?
b) Classify the random variable (finite discrete, infinite discrete, or continuous).
21. A random variable X is the number of hours a student practices probability
problems the day before the exam.
a) What values may X assume?
b) Classify the random variable (finite discrete, infinite discrete, or continuous).
22. The probability distribution of the random variable X, is represented by the
accompanying histogram, with the rectangle centered on the number 8 missing.
What should be the height of this rectangle?
Math 166 Week-in-Review - © S. Nite 10/6/2012
WIR #5
Page 5 of 5
23. Let X denote the random variable that gives the sum of the face that fall
uppermost when two fair dice are rolled. Find P(X = 8).
24. Three cards are drawn from a well-shuffled deck of 52 playing cards. Let X
denote the number of Kings drawn. Find P(X = 2). Round your answer to 3
decimal places.
25. The probability distribution of the random variable X is shown in the
accompany table.
x
P(X = x)
−2
0.16
-1
0.12
0
0.33
1
0.18
2
0.11
3
0.10
Find the following.
a) P(X ≤0)
b) P(X ≥3)
c) P(1 ≤X < 3)
d) P(X = 2)
e) P(X = 4)
26. A box has 5 yellow, 6 gray, and 3 black balls. If four balls are drawn without
replacement, and X is the number of gray balls drawn, compute the following:
a) P(X = 3)
b) P(X ≤ 3)
27. A baseball player’s probability of a hit when he comes up to bat is .35. Assume
a hit is independent of any previous at bat. Let X be the random variable given by
the number of hits in 4 successive at bats. Find the probability distribution.
28. A fair die is rolled 30 times. Find the probability of obtaining:
a) at most 5 rolls that are threes.
b) at least 10 rolls that are multiples of four.
29. An adverse reaction to a flu shot is 0.15. Flu shots are given to a group of 70
people. Find the probability that
a) 6 to 15 people will have an adverse reaction to the shot.
b) exactly 20 people will have an adverse reaction to the shot.
30. Draw the histogram for the probability distribution given below.
x
P(X = x)
3
0.3
5
0.2
7
0.15
9
0.25
11
0
13
0.1