This module introduces the concept of Probability, the chance of an event occurring.
By the end of this chapter, the student should be able to:
Understand and use the terminology of probability.
Determine whether two events are mutually exclusive and whether two events are independent.
Calculate probabilities using the Addition Rules and Multiplication Rules.
Construct and interpret Contingency Tables.
Construct and interpret Venn Diagrams (optional).
Construct and interpret Tree Diagrams (optional).
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn to solve probability problems using a systematic approach.
Your instructor will survey your class. Count the number of students in the class today.
Raise your hand if you have any change in your pocket or purse. Record the number of raised hands.
Raise your hand if you rode a bus within the past month. Record the number of raised hands.
Raise your hand if you answered "yes" to BOTH of the first two questions. Record the number of raised hands.
Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person in your class rode a bus within the last month and so on. Discuss your answers.
Find P(change).
Find P(bus).
Find P(change and bus) Find the probability that a randomly chosen student in your class has change in his/her pocket or purse and rode a bus within the last month.
Find P(change| bus) Find the probability that a randomly chosen student has change given that he/she rode a bus within the last month. Count all the students that rode a bus. From the group of students who rode a bus, count those who have change. The probability is equal to those who have change and rode a bus divided by those who rode a bus.
Probability: Terminology is part of the collection col10555 written by Barbara Illowsky and Susan Dean defines key terms related to Probability and has contributions from Roberta Bloom.
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin twice is an example of an experiment.
The result of an experiment is called an outcome. A sample space is a set of all possible outcomes. Three ways to represent a sample space are to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter S is used to denote the sample space. For example, if you flip one fair coin, S = {H, T} where H = heads and T = tails are the outcomes.
An event is any combination of outcomes. Upper case letters like A and B represent events. For example, if the experiment is to flip one fair coin, event A might be getting at most one head. The probability of an event A is written P(A).
The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between 0 and 1, inclusive (includes 0 and 1 and all numbers between these values). P(A) = 0 means the event A can never happen. P(A) = 1 means the event A always happens. P(A) = 0.5 means the event A is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative fequency of heads approaches 0.5 (the probability of heads).
Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head(H) and a Tail(T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.
To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the
total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the
sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four
outcomes. A = getting one head. There are two outcomes {HT, TH}. 
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Suppose you roll one fair six-sided die, with the numbers {1,2,3,4,5,6} on its faces. Let event E = rolling a number that is at least 5. There are two outcomes {5, 6}. 
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If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, 2/6 of the rolls would result in an outcome of "at least 5". You would not expect exactly 2/6. The long-term relative frequency of obtaining this result would approach the theoretical probability of 2/6 as the number of repetitions grows larger and larger.
This important characteristic of probability experiments is the known as the Law of Large Numbers: as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes don't happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.) The Law of Large Numbers will be discussed again in Chapter 7.
It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased . Two math professors in Europe had their statistics students test the Belgian 1 Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos have a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later in this chapter we will learn techniques to use to work with probabilities for events that are not equally likely.
An outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are NOT listed twice.
An outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}.
The complement of event A is denoted A' (read "A prime"). A' consists of all outcomes
that are NOT in A. Notice that P(A) + P(A') = 1. For example, let S = {1, 2, 3, 4, 5, 6}
and let A = {1, 2, 3, 4}. Then, 
The conditional probability of A given B is written P(A|B). P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A|B) is
P(A|B)=
where P(B) is greater than 0.
For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = face is 2 or 3 and B = face is even (2, 4, 6). To calculate P(A|B), we count the number of outcomes 2 or 3 in the sample space B = {2, 4, 6}. Then we divide that by the number of outcomes in B (and not S).
We get the same result by using the formula. Remember that S has 6 outcomes.
P(A|B)=
It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.
In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts (a) through (j) below. (Note that you can't find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)
Let F be the event that a student is female.
Let M be the event that a student is male.
Let S be the event that a student has short hair.
Let L be the event that a student has long hair.
| a. The probability that a student does not have long hair. |
| b. The probability that a student is male or has short hair. |
| c. The probability that a student is a female and has long hair. |
| d. The probability that a student is male, given that the student has long hair. |
| e. The probability that a student has long hair, given that the student is male. |
| f. Of all the female students, the probability that a student has short hair. |
| g. Of all students with long hair, the probability that a student is female. |
| h. The probability that a student is female or has long hair. |
| i. The probability that a randomly selected student is a male student with short hair. |
| j. The probability that a student is female. |
| a. P(L')=P(S) |
| b. P(M or S) |
| c. P(F and L) |
| d. P(M|L) |
| e. P(L|M) |
| f. P(S|F) |
| g. P(F|L) |
| h. P(F or L) |
| i. P(M and S) |
| j. P(F) |
**With contributions from Roberta Bloom
Probability: Independent and Mutually Exclusive Events is part of the collection col10555 written by Barbara Illowsky and Susan Dean and explains the concept of independent events, where the probability of event A does not have any effect on the probability of event B, and mutually exclusive events, where events A and B cannot occur at the same time. The module has contributions from Roberta Bloom.
Independent and mutually exclusive do not mean the same thing.
Two events are independent if the following are true:
P(A|B) = P(A)
P(B|A) = P(B)
P(A AND B) = P(A) ⋅ P(B)
Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are NOT independent, then we say that they are dependent.
Sampling may be done with replacement or without replacement.
With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
Without replacement:: When sampling is done without replacement, then each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.
If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.
A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0.
For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}. A AND B= {4, 5}. P(A AND B) = 
and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so
P(A AND C) = 0. Therefore, A and C are mutually exclusive.
If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise.
The following examples illustrate these definitions and terms.
Flip two fair coins. (This is an experiment.)
The sample space is {HH, HT, TH, TT} where T = tails and H = heads. The outcomes are HH, HT, TH, and TT. The outcomes HT and TH are different. The HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads.
Let A = the event of getting at most one tail. (At most one tail means 0 or 1 tail.) Then A can be written as {HH, HT, TH}. The outcome HH shows 0 tails. HT and TH each show 1 tail.
Let B = the event of getting all tails. B can be written as {TT}. B is the complement of A. So, B = A'. Also, P(A) + P(B) = P(A) + P(A') = 1.
The probabilities for A and for B are 
and