# Sequences, Probability, and Counting Theory

## Probability


### Learning Objectives

1. Introduction to Sample Spaces and Computing Basic Probabilities.


### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.


Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.


The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.

A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.

The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.

An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.

To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.

It is important to note that in order to use this formula, all outcomes must be equally likely to happen.

For example, the probability of rolling an even number with a standard dice is:


### Practice Makes Perfect

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

### Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is  An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event 
 is a number that always satisfies 
  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. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.

### Computing Probabilities of Equally Likely Outcomes

Let 
 be a sample space for an experiment. When investigating probability, an event is any subset of 
 When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in 
 Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in 
 so the probability of the event is 
 

### Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events 
  is the event that occurs if either or both events occur.
Suppose the spinner in  is spun. We want to find the probability of spinning orange or spinning a 
  
There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is 
  There are a total of 6 sections, and 2 of them have a 
 So the probability of spinning a 
 is 
  If we added these two probabilities, we would be counting the sector that is both orange and a 
 twice. To find the probability of spinning an orange or a 
 we need to subtract the probability that the sector is both orange and has a 
 
The probability of spinning orange or a 
 is 
 

### Computing the Probability of Mutually Exclusive Events

Suppose the spinner in  is spun again, but this time we are interested in the probability of spinning an orange or a 
 There are no sectors that are both orange and contain a 
 so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of 
 and 
 is the empty set. The probability of spinning an orange is 
 and the probability of spinning a 
 is 
 We can find the probability of spinning an orange or a 
 simply by adding the two probabilities.
The probability of spinning an orange or a 
 is 
 

### Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event 
 denoted 
  is the set of outcomes in the sample space that are not in 
 For example, suppose we are interested in the probability that a horse will lose a race. If event 
 is the horse winning the race, then the complement of event 
 is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is 
 the probability of the horse losing the race is simply

### Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are 
 ways to select 2 phones that are not defective. There are 8 phones, so there are 
 ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

### Key Equations


### Key Concepts

1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .

### Section Exercises


### Verbal


### Numeric

For the following exercises, use the spinner shown in  to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of 
  cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains 
  blue, 
  brown, 
  orange, 
  yellow, 
  red, and 
  green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

### Extensions

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 
  numbers from the numbers 
  to 
  After the player makes his selections, 
  winning numbers are randomly selected from numbers 
  to 
  A win occurs if the player has correctly selected 
  or 
  of the 
  winning numbers. (Round all answers to the nearest hundredth of a percent.)

### Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

### Chapter Review Exercises


### Sequences and Their Notation


### Arithmetic Sequences


### Geometric Sequences


### Series and Their Notation


### Counting Principles


### Binomial Theorem


### Probability

For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

### Practice Test

For the following exercises, use the spinner in .