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In this lesson, you will study the difference between dependent and independent events and how to use conditional probability.
Sometimes the probability of an event can change, given the number of previous trials or information about other events. Two events are independent if the probability of the second event is not affected by the outcome of the first event. If, instead, the outcome of the first event does affect the probability of the second event, these events are dependent.
Examples of independent events:
Examples of dependent events:
Imagine a bag containing 5 red marbles, 3 white marbles, and 2 blue marbles. We pick two marbles from the bag, first one and then the other, trying to get two of the same color. This scenario can produce either independent or dependent events.
If we replace the marble we chose first and mix the marbles before making our second choice, then the second choice will be independent from the first. However, if we do not replace the first marble before choosing the second, this alters the probability of the second pick by changing the number of possible outcomes: there is one fewer possibility.
When computing the probability of two events, it is important to note whether the events are independent or dependent. The event’s status can change depending on the situation.
Consider the bag of marbles above. What is the probability of choosing two white marbles, one after the other, if:
(a) we replace the first marble before choosing a second marble?
(b) we do not replace the first marble before choosing a second marble?
Let A = “the first marble is white” and B = “the second marble is white.”
Case (a): in which the first marble is replaced—is an example of two independent events. Notice that 
, because the two events occur under the exact same circumstances.
Case (b)—in which the first marble is not replaced—is an example of a dependent event. If we pick a marble and do not replace it, the sample space has changed. We still have 
. However, for the second drawing, the sample space consists of only 9 marbles and, assuming that A was successful, there are only 2 white marbles left in the bag. Thus, 
and 
. Thus we know that case (a) is more probable than case (b).
When calculating the probability of dependent events, we can simplify matters by assuming that the first event resulted in the desired outcome when computing the probability of the second event. Keep this in mind when calculating dependent probabilities.
For example, when computing probabilities of various choices from the bag of marbles, we assumed that event A was successful (had a favorable outcome, i.e., that a white marble was drawn). Then 2 white marbles would remain in the bag, and the probability of drawing another is 
.
If we know that an event has already occurred and we know its outcome, how does this alter the probability of another event’s outcome?
The answer to this question depends on whether the two events are dependent or independent. In general, conditional probability measures the probability of an outcome B provided we know that another outcome A has already occurred.
The formula for conditional probability is 
.
The idea of conditional probability is closely related to the difference between the probability of dependent events and the probability of independent events. Recall that when computing the probability of two dependent events A and B, we can assume that A was successful to simplify computing the probability of B. Because we make this assumption, we really compute is 
. We can rearrange this formula to get the equation we will use for conditional probability.
If the two events are independent then it does not matter whether A has occurred, because A has no influence on the probability of B. Thus 
.
If A and B are the outcomes of dependent events, then
and thus 
If A and B are the outcomes of independent events, then
and 
When we need to consider more than two events, we can use induction based on what we already know about two events. Suppose we have three events and three favorable outcomes A, B, and C. We want to compute 
. To make sense of this scenario, we can rewrite it and formulate the result using what we know about two events. Say 
.
If A, B, and C are all independent, then
If A, B, and C are all dependent, then
Suppose we choose two cards from a deck of 52, making our choices one at a time without replacing cards. What is the probability of the first card being a black face card and the second card being a heart?
The correct choice is A. Let A = “the first card drawn is a black face card” and B = “the second card drawn is a heart.” Because we did not replace the first card we chose, these two events are dependent, and thus 
The local bookstore often gets new shipments of math books. In fact, the probability that they will receive new math books on any given day is 76%. If a shipment of new math books does arrive, you expect that there is a 30% probability that the particular book you want to buy will be in that shipment. What is the probability that there will be a shipment of new math books today and that the book you want to buy will be in that shipment?
|
A
|
22.8% |
|
B
|
25% |
|
C
|
26.2% |
|
D
|
28% |
The correct choice is A. Let A = “a shipment of new math books arrives today” and B = “the book you want to buy is in that shipment.” The question asks us to determine the probability that new math books arrive and that the book you want to buy is among them, in other words, we want 
.
We know that 
. Given the events described above, we get 
= “provided the bookstore receives new math books today, your book will be among them.” The question tells us that 
. Therefore, using our equation for conditional probability, we get
When solving probability questions, it helps to keep track of the events and their outcomes. To ensure that this information is clear, in the very beginning, always define the outcomes of the events with which you are working . Write them down so that you can refer back to your notes if you get confused.
There are 75 students in a freshman chemistry class at university Q. Before the first exam, the students answered a questionnaire about whether they had studied and the following table was compiled from their responses and their grades.
| Chemistry Class | ||
|---|---|---|
|
Students who studied
|
Students who didn’t study
|
|
| # of students who passed |
55
|
6
|
| # of students who failed |
2
|
12
|
What is the probability that a random student who did not study passed the exam?
The correct choice is B. Let A = “this student did not study” and B = “this student passed the test.” We want to find 
, and to find this quantity we will use 
.
Therefore the probability that a random student who did not study passed the test is 
.
.
is used to count the elements of the sample space when ordering r elements out of a set of n elements and order does matter. 
is used to count the elements of the sample space when grouping r elements out of a set of n elements and order does not matter. 
.