A or Bor
$A∪B.$
For two mutually exclusive events A and B, the probability that A or B occur in one trial is the sum of the individual probability of each event.
For example, consider rolling a standard six-sided die. Let A be the event that an even number is rolled and B be the event that a prime number is rolled.
Event | Outcome(s) | Probability |
---|---|---|
Even | 2, 4, 6 | $P(A)=63 =21 $ |
Prime | 2, 3, 5 | $P(B)=63 =21 $ |
Even and prime | 2 | $P(AandB)=61 $ |
For mutually exclusive events, the Addition Rule of Probability is a postulate.
Therefore, no proof will be given for mutually exclusive events. Now, consider non-mutually exclusive events A and B.
In the Venn diagram above, it can be seen part of event A does not overlap event B. That part is labeled a. Similarly, the part of event B that does not overlap event A is labeled b. Furthermore, the overlapping part – also known as the intersection — of both events is labeled c.Notation | Meaning |
---|---|
P(A)=a+c | The probability of A happening is a+c. |
P(B)=b+c | The probability of B happening is b+c. |
$P(A∪B)=a+b+c$ | The probability of A happening or B happening is a+b+c. |
Identity Property of Addition
Rewrite 0 as c−c
Commutative Property of Addition
Associative Property of Addition
Substitute values
Two events A and B are independent events if the occurrence of either of these events does not affect the occurrence of the other. It is also said that they are independent if and only if the probability that both events occur is equal to the product of the individual probabilities.
For example, consider drawing two marbles from a bowl, one at a time.
Two events A and B are considered dependent events if the occurrence of either of these events affects the occurrence of the other. If the events are dependent, the probability that both events occur is equal to the product of the probability of the first event occurring and the probability of the second event occurring after the first event.
For example, consider drawing two marbles from a bowl, one at a time.
Conditional probability is the measure of the likelihood of an event B occurring, given that event A has occurred previously. The probability of B given A is written as P(B∣A). It can be calculated by dividing the probability of the intersection of A and B by the probability of A.
$P(B∣A)=P(A)P(AandB) ,whereP(A) =0$
The intuition behind the formula can be visualized by using Venn Diagrams. Consider a sample space S and the events A and B such that P(A)≠0.
Assuming that event A has occurred, the sample space is reduced to A.
This means that the probability that event B can happen is reduced to the outcomes in the intersection of A and B, that is, to those outcomes in $A∩B.$
The possible outcomes are given by P(A) and the favorable outcomes by $P(A∩B).$ Therefore, the conditional probability formula can be obtained using the probability formula.
$P(B∣A)=P(A)P(AandB) $
A and Bor
$A∩B.$
For two independent events A and B, the probability that the intersection of A and B occurs is the product of the individual probabilities.
P(A and B)=P(A)⋅P(B)
Conversely, if A and B are dependent events, a rearrangement of the Conditional Probability Formula can be used to find the probability of the intersection of the events.
Consider a box with $four$ red marbles and $six$ blue marbles. In an experiment, two marbles are drawn randomly from the box without replacement. The Multiplication Rule of Probability can be used to find the probability that both marbles are red.
A two-way frequency table, also known as a two-way table, is a table that displays categorical data that can be grouped into two categories. One of the categories is represented by the rows of the table, the other by the columns. For example, the table below shows the results of a survey in which 100 participants were asked if they have a driver's license and if they own a car.
Here, the two categories are car
and driver's license,
both with possible answers of yes
and no.
The entries in the table are called joint frequencies. Two-way frequency tables often include the total of the rows and columns. These totals are called marginal frequencies.
Totalrow and the
Totalcolumn is equal to the sum of all joint frequencies and is called the grand total. In the case of the survey, the grand total is 100. From the table it can be read that, among other things, 43 people both have a driver's license and own a car. It can also be read that 33 people do not have a driver's license.
Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. To draw a two-way frequency table, three steps must be followed.
Suppose that 53 people took part in an online survey, where they were asked whether they prefer top hats or berets. Out of the 18 males that participated, 12 of them prefer berets. Also, 15 of the females chose top hats as their preference. The steps listed above will be developed for this example.
First, the two categories of the table must be determined, after which the table can be drawn without frequencies. Here, the participants gave their hat preference and their gender, which are the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male.
The total row and total column are included to write the marginal frequencies.
The given joint and marginal frequencies can now be added to the table.
In a two-way frequency table, a joint relative frequency is the ratio of a joint frequency to the grand total. Similarly, a marginal relative frequency is the ratio of a marginal frequency to the grand total. Consider an example two-way table.
Here, the grand total is 100. The joint and marginal frequencies can now be divided by 100 to obtain the $joint$ and $marginal$ relative frequencies. Clicking in each cell will display its interpretation.
A conditional relative frequency is the ratio of a joint frequency to either of its corresponding two marginal frequencies. Alternatively, it can be calculated using joint and marginal relative frequencies. As an example, the following data will be used.
Using the column totals, the left column of joint frequencies should be divided by 67, and the right column by 33. Since the column totals are used, the sum of the conditional relative frequencies of each column is 1.
The resulting two-way frequency table can be interpreted to obtain the following information.
From the two-way frequency table, find the conditional relative frequencies based on the columns. Then, find the probability that a vegetarian has a pet.
Vegetarian | ||||
Yes | No | Total | ||
Pet | Yes | 0.456 | 0.154 | 0.61 |
No | 0.123 | 0.267 | 0.39 | |
Total | 0.579 | 0.421 | 1 |
To begin, recall that conditional relative frequencies can be calculated by dividing the joint relative frequencies by the marginal relative frequencies. Since it should be based on the columns, it's the totals of the vegetarians, 0.579 and 0.421, that are used as the denominators.
Vegetarian | |||
Yes | No | ||
Pet | Yes | $0.5790.456 ≈0.79$ | $0.4210.154 ≈0.37$ |
No | $0.5790.123 ≈0.21$ | $0.4210.267 ≈0.63$ | |
Total | 0.579 | 0.421 |
Note that the sum of the conditional relative frequencies in each column is equal to 1. Now, to find the probability that a vegetarian has a pet, we look at the column for people who answered "Yes" on "Vegetarian".
Vegetarian | |||
Yes | No | ||
Pet | Yes | 0.79 | 0.37 |
No | 0.21 | 0.63 | |
Total | 1 | 1 |
In that column, $79%$ said "Yes" to having a pet and $21%$ said "no". Thus, the probability that a vegetarian has a pet is $79%.$