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Probability and Counting Principles
College Algebra
Counting Principles
According to the Addition Principle, if one event can occur in 𝑚 ways and a second event with no common outcomes can occur in 𝑛 ways, then the first or second event can occur in 𝑚 + 𝑛 ways.
According to the Multiplication Principle, if one event can occur in 𝑚 ways and a second event can occur in 𝑛 ways after the first event has occurred, then the two events can occur in 𝑚×𝑛 ways. This is also known as the Fundamental Counting Principle.
Example: On a restaurant menu, there are 3 appetizer options, 2 vegetarian entrée options and 5 meat entrée options, and 2 dessert options.There are 3× 2 + 5 ×2 = 42 different choices for a three-course dinner.
Permutations of Distinct Objects
Given 𝑛 distinct objects, the number of ways to select 𝑟 objects from the set in order is:
𝑃 𝑛, 𝑟 =𝑛!
𝑛 − 𝑟 !
Example: A club with six people need to elect a president, a vice president and a treasurer. How many ways can the officers be elected?
Solution: 𝑃 6,3 = 0!012 !
= 0343532363723637
= 6 3 5 3 4 = 120
Permutations of Non-Distinct Objects
If there are 𝑛 elements in a set and 𝑟7 are alike, 𝑟6 are alike, 𝑟2 are alike, and so on through 𝑟:, the number of permutations can be found by:
𝑛!𝑟7! 𝑟6!⋯ 𝑟:!
Example: Find the number of rearrangements of the letters in the word DISTINCT.Solution: There are 8 letters in the word, but I and T are repeated two times each. Therefore, the number of permutations is:
8!2! 3 2! =
8 3 7 3 6 3 5 3 4 3 3 3 2 3 12 3 2 = 10080
Combinations
When we are selecting objects and the order does not matter, we are dealing with combinations.Given 𝑛 distinct objects, the number of ways to select 𝑟 objects from the set is:
𝐶 𝑛, 𝑟 =𝑛!
𝑟! 𝑛 − 𝑟 !
Example: An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?
Solution: 𝐶 10,3 = 7?!2! 7?12 !
= 7?!2!@!
= 7?3A3B3@!2363@!
= @6?0= 120
Number of Subsets of a Set
A set containing 𝑛 distinct objects has 2C subsets.
Example: A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a baked potato?
Solution: There are 4 options, so there are 25 = 16 possible ways to order a baked potato.This result is the same as:
𝐶 4,0 + 𝐶 4,1 + 𝐶 4,2 + 𝐶 4,3 + 𝐶 4,4 = 1 + 4 + 6 + 4 + 1 = 16
Binomial Coefficients
In the shortcut to finding 𝑥 + 𝑦 C we use combinations to find the coefficients that will appear in the expansion of the binomial.If 𝑛 and 𝑟 are positive integers with 𝑛 ≥ 𝑟, then the binomial coefficientis:
𝑛𝑟 = 𝐶 𝑛, 𝑟 =
𝑛!𝑟! 𝑛 − 𝑟 !
Note that CG = C
C1G .
Example: A6 = A!
6! A16 != A!
6!@!= A3B3@!
63@!= 36
A@ = A!
@! A1@ != A!
@!6!= A3B3@!
@!36= 36
Binomial Theorem
The Binomial Theorem is a formula that can be used to expand any binomial.
𝑥 + 𝑦 C = ∑ C: 𝑥
C1:𝑦:C:I?
= 𝑥C + C7 𝑥
C17𝑦 + C6 𝑥
C17𝑦6 + ⋯+ CC17 𝑥𝑦
C17 + 𝑦C
Example: Expand 𝑥 + 𝑦 4
=50 𝑥4𝑦? +
51 𝑥5𝑦7 +
52 𝑥2𝑦6 +
53 𝑥6𝑦2 +
54 𝑥7𝑦5 +
55 𝑥?𝑦4
= 𝑥4 + 5𝑥5𝑦 + 10𝑥2𝑦6 + 10𝑥6𝑦2 + 5𝑥𝑦5 + 𝑦4
Using the Binomial Theorem to Find a Single Term
The (𝑟 + 1)th term of the binomial expansion of (𝑥 + 𝑦)C is:𝑛𝑟 𝑥C1G𝑦G
Example: Find the sixth term of 3𝑥 − 𝑦 A without fully expanding the binomial.Solution:Let 𝑟 = 5 for the sixth term, and use 3𝑥 and −𝑦 for the two variables.
95 3𝑥 A14 −𝑦 4 =
9 3 8 3 7 3 6 3 5!4! 3 5! 35𝑥5(−1)4𝑦4 = −10206𝑥5𝑦4
Probabilities
The likelihood of an event is known as probability. The probability of an event 𝑝 is a number that always satisfies 0 ≤ 𝑝 ≤ 1, where 0 indicates an impossible event and 1 indicates a certain event.
A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities.
The probability of an event 𝐸 in an experiment with sample space 𝑆 with equally likely outcomes is given by
𝑃 𝐸 =numberofelementsin𝐸numberofelementsin𝑆 =
𝑛(𝐸)𝑛(𝑆)
𝐸 is a subset of 𝑆, so it is always true that 0 ≤ 𝑃(𝐸) ≤ 1.
Probability for Multiple Events
The probability of the union of two events 𝐸 and 𝐹 (written 𝐸 ∪ 𝐹) equals the sum of the probability of 𝐸 and the probability of 𝐹 minus the probability of 𝐸 and 𝐹 occurring together (which is called the intersection of 𝐸and 𝐹 and is written as 𝐸 ∩ 𝐹).
𝑃 𝐸 ∪ 𝐹 = 𝑃 𝐸 + 𝑃 𝐹 − 𝑃(𝐸 ∩ 𝐹)
Example: A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.
Solution: 𝑃 ℎ = 7246
, 𝑃 7 = 546
, and 𝑃 ℎ ∩ 7 = 746
𝑃 ℎ ∪ 7 = 7246+ 546− 746= 70
46= 5
72
Computing the Probability of Mutually Exclusive Events
The probability of the union of two mutually exclusive events 𝐸 and 𝐹 is given by
𝑃 𝐸 ∪ 𝐹 = 𝑃 𝐸 + 𝑃 𝐹
Example: A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.
The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is 7
5, and the probability of drawing a spade is also 7
5, so the probability
of drawing a heart or a spade is 75+ 75= 7
6
Probability That an Event Will Not Happen
The complement of an event 𝐸, denoted 𝐸′, is the set of outcomes in the sample space that are not in 𝐸.
𝑃 𝐸c = 1 − 𝑃(𝐸)
Example: Two six-sided dice are rolled. What is the probability that the sum of the numbers is greater than 3?Solution:The sample space is the set of all 36 possible outcomes from 1 + 1 to 6 + 6. It is easier to consider the 3 possible totals not greater than 3: 1 + 1, 1 + 2, and 2 + 1. Therefore, if 𝑃 𝐸c = 2
20= 7
76, then 𝑃 𝐸 = 77
76.
Computing Probability Using Counting TheoryMany probability problems use permutations and combinations to find the number of elements in events and sample spaces.
Example: A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears. Find the probability that 2 bears and 3 dogs are chosen.
Solution:There are 6 bears, so there are 𝐶(6,2) ways to choose 2 bears.There are 5 dogs, so there are 𝐶(5,3) ways to choose 3 dogs.There are 𝐶(6,2) 3 𝐶(5,3) ways to choose 2 bears and 3 dogs.There are 14 toys, so there are 𝐶 14,5 ways to choose 5 toys.
𝑃 𝐸 =𝐶(6,2) 3 𝐶(5,3)
𝐶(14,5) =15 3 102002 =
751001
Quick Review
• What is the Fundamental Counting Principle?• What is the difference between a permutation and a combination?• What is the formula for the number of permutations of non-distinct items?• How many subsets are there for a set of 𝑛 distinct items?• What is the formula for a binomial coefficient?• How is the Binomial Theorem used?• What is the sum of probabilities for all possible events in a given
probability model?• How do you compute the probability of the union of two events?
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