In this Concept, you will learn about the concept of conditional probability and be presented with some examples of how conditional probability is used in the real world. You will also learn the appropriate notation associated with conditional probability.
Watch This
For an introduction to conditional probability (2.0), see SomaliNew, Conditonal Probability Venn Diagram (4:25).
For an explanation of how to find the probability of "And" statements and dependent events (2.0), see patrickJMT, Calculating Probability  "And" Statements, Dependent Events (5:36).
Guidance
Notation
We know that the probability of observing an even number on a throw of a die is 0.5. Let the event of observing an even number be event
The only even number in the sample space for
Conditional Probability of Two Events
If
To calculate the conditional probability that event
For our example above, the die toss experiment, we proceed as is shown below:
To find the conditional probability, we use the formula as follows:
A medical research center is conducting experiments to examine the relationship between cigarette smoking and cancer in a particular city in the USA. Let
Simple Events  Probabilities 


0.10 

0.30 

0.05 

0.55 
Figure: A table of probabilities for combinations of smoking,
These simple events can be studied, along with their associated probabilities, to examine the relationship between smoking and cancer.
Example A
Determine the rates of developing cancer for smokers and nonsmokers.
Solution:
First, let's write out events symbolically:
A very powerful way of examining the relationship between cigarette smoking and cancer is to compare the conditional probability that an individual gets cancer, given that he/she smokes, with the conditional probability that an individual gets cancer, given that he/she does not smoke. In other words, we want to compare
Recall that
Before we can use this relationship, we need to calculate the value of the denominator.
This tells us that according to this study, the probability of finding a smoker selected at random from the sample space (the city) is 40%. We can continue on with our calculations as follows:
Similarly, we can calculate the conditional probability of a nonsmoker developing cancer:
In this calculation,
Example B
Use the calculations from Example A to determine how many more times likely a smoker is to develop cancer than a nonsmoker is.
Solution:
From the calculations in Example A, we can clearly see that a relationship exists between smoking and cancer. The probability that a smoker develops cancer is 25%, and the probability that a nonsmoker develops cancer is only 8%. The ratio between the two probabilities is
Natural Frequencies Approach
There is another and interesting way to analyze this problem, which has been called the natural frequencies approach (see G. Gigerenzer, “Calculated Risks” Simon and Schuster, 2002).
Example C
Use the natural frequencies approach to find the probability of having cancer given that you smoke.
Solution:
We will use the probability information given above to demonstrate this approach. Suppose you have 1000 people. Of these 1000 people, 100 smoke and have cancer, and 300 smoke and don’t have cancer. Therefore, of the 400 people who smoke, 100 have cancer. The probability of having cancer, given that you smoke, is
Of these 1000 people, 50 don’t smoke and have cancer, and 550 don’t smoke and don’t have cancer. Thus, of the 600 people who don’t smoke, 50 have cancer. Therefore, the probability of having cancer, given that you don’t smoke, is
Guided Practice
In a recent election, 35% of the voters were democrats and 65% were not. Of the democrats, 75% voted for candidate Z and of the nonDemocrats, 15% voted for candidate Z. Define the following events:
A = voter is Democrat
B = voted for candidate Z
a. Find
b. Find
c. Find
d. Find
Solutions:
a.
b.
c.
d.
Explore More
For 15, two fair coins are tossed.
i. List all the possible outcomes in the sample space.
ii. Suppose two events are defined as follows:
Find the probabilities:

P(A) 
P(B) 
P(A∩B) 
P(AB) 
P(BA)
For 611, a box of six marbles contains two white marbles, two red marbles, and two blue marbles. Two marbles are randomly selected without replacement, and their colors are recorded.
i. List all the possible outcomes in the sample space.
ii. Suppose three events are defined as follows:
Find the probabilities:

P(BA) 
P(BA′) 
P(BC) 
P(AC) 
P(CA′)  If
P(A)=0.3 ,P(B)=0.7 , andP(A∩B)=0.15 , findP(AB) andP(BA) .  In a large class, 65% of the students are liberal arts majors and 35% are science majors. Twentyfive percent of the liberal arts majors are seniors while 45% of the science majors are seniors.
 If there are 200 students in the class, how many of them are science majors?
 If there are 200 students in the class, how many of them are science majors and seniors?
 Create a twoway table with the row variable as the major (science or liberal arts) and the column variable as the class (senior, nonsenior)
 What percent of the class are seniors?
 Make a tree diagram for this situation.
 Use the tree diagram to determine the percentage of the class that is seniors.
 In a restaurant, a waitress notices that 75% of her customers order coffee and that 30% of her customers order coffee and a croissant. What is the probability that a given customer would order a croissant, given that he/she has ordered coffee?
 Suppose a middle school has grades 7 and 8 with the same number of students in each grade. Half of the students in grade 8 are taking algebra 1 and 25% of the students in grade 7 are taking algebra 1. Suppose that an algebra 1 student is randomly chosen to attend a math competition with a math teacher. What is the probability that the student is in grade 7?
 If A and B are independent, determine the probability of each of the following:
 Both A and B
 Either A or B
 Neither A nor B
 A but not B
 A given that B occurs
 In a class has 30 students, 15 play tennis, 19 play volleyball and 2 play neither of these sports. A student is randomly selected from the class. Determine the probability that the student:
 Plays both tennis and volleyball
 Plays at least one of these two sports
 Plays volleyball given that he/she does not play tennis.
 According to the Migration Information Source and the U.S. Census Bureau, “only 35 percent of the foreignborn people in the US in 1997 were naturalized citizens, compared with 42.5% in 2007. What is the probability that two randomly selected foreignborn people in the US in 2007 were both naturalized citizens?
 65% of the students in a high school are female. Of the male students 11% are colorblind and of the female students 4% are colorblind. If a randomly chosen student
 is colorblind, find the probability that the student is female.
 Is not colorblind, find the probability that the student is male.
 A jar contains 5 red and 4 nonred marbles. One marble is randomly drawn without replacement from the jar and its color is noted. A second marble is then drawn. What is the probability that:
 The second marble is red?
 The first was nonred given that the second was red?
 Suppose that 5% of men are colorblind and 25% of women are colorblind. A person is chosen at random and that person is colorblind. What is the probability that the person is male? (Assume males and females are in equal numbers).
Keywords
Event
Factorial
Intersection of events
Mutually exclusive
Sample space