M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
2.1 Multiplication Principle and Permutations
Multiplication Principle : Suppose there are m ways of performing task T1, and n ways of performing
task T2. Then, there are mn ways of performing the task T1 followed by task T2.
Example 1. A card is picked from a standard deck of 52 cards and then a 6-sided die is rolled. How
many possible outcomes are there?
Example 2.There are 3 ways to go from Point A to Point B, and 2 ways to go from Point B and Point
C. Find the number of ways of going from Point A to Point C, via Point B.
Permutations
Calculator Functions:
1. Factorial :
MATH → scroll to PRB → scroll to 4 : ! → ENTER
Example: Find 5! .
Enter 5 → MATH → scroll to PRB → scroll to 4 : ! → ENTER
2. Permutation : nPr
Enter n → MATH → scroll to PRB → scroll to 2 : nPr → Enter r → ENTER
Example: Find P(5 , 2) .
Here, n = 5 and r = 2.
Enter 5 → MATH → scroll to PRB → scroll to 2 : nPr → Enter 2 → ENTER
Example 3.In how many different ways can the digits in the set S = {1, 2, 3} be arranged?
Example 4. How many ways can 10 kids be lined up to form a queue?
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Example 5. At an award ceremony, 5 couples are to be called one at a time to receive an award. In how
many ways can this be done if
a) men and women must alternate?
b) Couples are called together ?
Example 6. A password can be formed with 4 letters and 5 digits. How many different passwords are
possible if
a) repetition of digits and/or numbers is allowed?
b) repetition of letters is not allowed?
c) repetition of digits is not allowed?
d) repetition is not allowed and the first digit must be non-zero ?
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Example 7.Ted loves traveling. He wants to visit 7 cities in Great Britain, 5 in France, 3 in Germany
and 4 in India. How many ways can his itinerary be made out?
Example 8.An exam consists of 5 true/false questions followed by 3 multiple choice questions each
with 4 answers. How many ways can a student answer all the questions if
a) he/she has to answer all the questions?
b) he/she is allowed to leave the questions blank?
Example 9. 10 friends decide to take a road-trip and rent a mini van. How many seating arrangements
are possible if only 4 of them know how to drive?
Example 10. Linda has 40 books, of which 17 are fiction, 6 are self – help, 13 are textbooks, and 4 are
cook-books. None of the books are exactly alike. How many ways can she arrange all the books on a
shelf if
a) there are no restrictions on the arrangement ?
b) books of the same category must be placed together ?
c) there's room for only 25 books ?
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Permutations of n Objects, Not All Distinct : Given a set of n objects in which n1 are alike and of one
kind, n2 objects are alike and of another kind, .., and nm objects are alike and of yet another kind, so that
n1 + n2 + ...+ nm = n, then,
The number of permutations of these n objects taken n at a time is given by
Example 11.How many different (distinguishable) ways can we arrange the letters of the word
MISSISSIPPI.
Example 12.If there are 3 red, 5 green, and 4 blue balls, how many ways can the balls be arranged,
given the balls of same color are exactly alike?
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
2.2 and 2.3 Combinations and Probability Applications of Counting
Calculator Functions:. Combination : nCr
Enter n → MATH → scroll to PRB → scroll to 3 : nCr → Enter r → ENTER
Example: Find C(5 , 2) .
Here, n = 5 and r = 2.
Enter 5 → MATH → scroll to PRB → scroll to 3 : nCr → Enter 2 → ENTER
Example 13. Out of a total of 10 people, a committee needs to be formed with a President, a
Vice-President, a Secretary, and 3 members. How many ways can the selection be made?
Example 14.A committee of 15 people consists of 8 men and 7 women. Find (i) the number of ways
and (ii) the probability, of choosing a subcommittee of 5 people consisting of
(i) Number of ways
a)
any 5 committee members
b)
all men
c)
exactly 3 men
d)
at least
e)
at most 3 men
(ii) Probability
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Example 15. An urn contains 5 green, 4 red, and 3 blue balls. Find (i) the number of ways and (ii) the
probability of choosing,
(i) Number of ways
a)
7 balls
b)
7 balls, with exactly 2 red
c)
7 balls, with at least 2 red
d)
7 balls, with no red
e)
7 balls, such that exactly 3 are
of the same color
f)
7 balls, with exactly 3 red and 2
green
g)
7 balls, with exactly 3 red or 2
green
h)
7 balls, with exactly 4 red or 4
blue
(ii) Probability
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
2.4 Bernoulli Trials
A Bernoulli trial is one which has only two possible outcomes – success and failure. If probability of
success is p, then the probability of failure is (1 – p).
A Binomial experiment is one where Bernoulli trials are repeated such that
a) The trials are repeated a finite number of times, n.
b) Each trial is independent of any other trial.
c) Each trial has only two possible outcomes – success and failure.
d) Probability of success for each trial is p
Calculator Functions – Binomial Trials:
a) binompdf – Exactly k successes
2nd VARS → scroll down to A : binompdf( → Enter n → Press the Comma key → Enter p → Press
the Comma key → Enter k → ENTER
where n : Total number of trials ; p : probability of a success ; k : number of successes
b) binomcdf – at most k successes
2nd VARS → scroll down to B : binomcdf( → Enter n → Press the Comma key → Enter p → Press
the Comma key → Enter k → ENTER
Note : “binomcdf” function is also used to calculate “at least” k successes. We'll discuss in class
how that's done.
Example 16.An unfair coin is flipped 8 times in succession. The probability of getting a head is 1/3. What
is the probability of getting at least two heads?
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Example 17. An unfair coin is flipped 120 times in succession. The probability of getting a heads is 0.4.
What is the probability of getting
a) exactly 40 heads
b) exactly 40 tails
c) At most 40 heads
d) at least 40 heads
e) More than 40 but less than 60 heads
Example 18. A machine produces defective parts 20% of the time. What is the probability that, of the
next 7 parts,
a) Exactly 3 are defective
b) At most 3 are defective
c) At least 3 are defective
M166
Ch 2 – Counting and Probability
TAMU – Spring 2016
Example 19. Your quiz has 10 multiple-choice questions, each with 4 choices and only one correct
answer. You guess on each question. What is the probability of you getting exactly 7 questions correct?
Example 20. A stock of 90 transistors is to be shipped. It turns out that 5 of the transistors are
defective.
(a) If a person selects 3 transistors at random from the stock, what is the probability that they are all
defective?
(b) If a person selects 3 cartridges at random from the stock, what is the probability that at least 1 is
defective?