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Probability

Finding the Probability of Two Events

Consider two events and The probability that or will occur is the probability of the union of and and can be found using the Addition Rule of Probability.

Rule

Addition Rule of Probability

For two mutually exclusive events  and the probability that  or occur in one trial is the sum of the individual probability of each event.

For example, consider rolling a standard six-sided die. Let be the event that a is rolled and be the event that a is rolled. The probability of or can be found by adding the individual probabilities. The formula above can be generalized to events that are not necessarily mutually exclusive. If events are overlapping, the probability of the common outcomes are counted twice in so an adjustment is needed.

For example, consider rolling a standard six-sided die. Let be the event that an even number is rolled and be the event that a prime number is rolled.

Event Outcome(s) Probability
even
prime
even and prime

Using the formula gives the probability that the result of the roll is even or prime.

This can be verified by noticing that there are five outcomes that are even or prime, and
Concept

Independent and Dependent Events

Sometimes, the occurrence of one event affects the occurrence of another. If this is the case, the events are said to be dependent. If not, they are independent.
Rule

Independent Events

Two events, and are independent if and only if the probability that both events occur is equal to the product of the individual probabilities.

If a coin is flipped two times, the outcome of the first flip does not affect the outcome of the second flip. For example, suppose the first flip is heads. This does not affect the likelihood that the second flip is also heads. By showing that the expressions are equal, it can be concluded that the events are independent. To find the probability of flipping heads twice a tree diagram can be drawn.

The number of possible outcomes when flipping two coins is Additionally, the favorable outcome, two heads, is Next, consider flipping a coin two separate times. The probability of flipping heads is

The probability is the same for both expressions, Therefore, because the rule is satisfied, the events are independent.
Concept

Dependent Events

Two events are said to be dependent when the occurrence of one affects the occurrence of the other. For example, consider drawing two marbles from a bowl, one at a time.

Bowl with marbles.svg

The probability of first picking a green marble can be calculated by dividing the favorable outcomes by the possible outcomes. There is green marble and total marbles. Suppose that the first marble is replaced before the second draw. Therefore, after the replacement, there is purple marble and total marbles. The combined probability of picking a green marble first and a purple marble second can be calculated using the Multiplication Rule of Probability. These events are not dependent. Suppose instead that, after the green marble is picked, it is not replaced in the bowl.

Bowl with red and purple marble.svg

This affects the probability of picking a purple marble on the second draw. Now, there still is purple marble but, instead of there are total marbles. With this information, the probability of picking green and then purple can be calculated.

As can be seen, these events are dependent because the occurrence of the first affects the occurrence of the second.
Rule

Conditional Probability

The conditional probability of an event is the probability that will occur given that another event has already occurred. The probability of given is written It can be calculated by dividing the probability for and with the probability of

Notice that is an intersection and can be calculated using the Multiplication Rule of Probability.
Concept

Intersection - Probability

Consider two events and The probability that and will occur is the probability of the intersection of and and can be calculated using the Multiplication Rule of Probability.

Rule

Multiplication Rule of Probability

For two independent events and the probability that and occur is the product of the individual probabilities.

For example, when rolling two dice, the probability of rolling two even numbers can be calculated using the individual probabilities. Note that there are outcomes that are even — and The probability of rolling two even numbers can be calculated using the formula. For not necessarily independent events, the conditional probability formula can be rearranged to a product form.

For example, consider a box with four red and six blue marbles. In an experiment two marbles are drawn randomly from the box. The formula can be used to find the probability that both marbles are red. Let and be the events that the first and second marbles are red.

  • The probability that the first marble is red is
  • Once a red marble is picked, the box only has three red and six blue marbles. The probability that the next marble is red is

The formula gives the probability that both marbles are red.

Concept

Two-Way Frequency Table

When categorical data belongs to two categories, such as if people are asked whether they own a car and whether they have a driver's license, it can be presented in a two-way frequency table. One of the categories is represented by the rows of the table, and the other by the columns. The above survey, with participants, could result in the following answers.

Driver's license
Yes No
Car Yes
No

The two categories are then "car" and "driver's license," both with the possible answers "yes" and "no." The entries in the table are called joint frequencies. Often, two-way frequency tables include the total of the rows and columns. These totals are called marginal frequencies. The sum of the "total" row and "total" column are each equal to the sum of all joint frequencies, in this case.

Driver's license
Yes No Total
Car Yes
No
Total
From the table, it can, for instance, be read that out of the people both own a car and have a driver's license, and that of the do not have a driver's license.
Method

Drawing a Two-Way Frequency Table

Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. Consider the following survey.

people took part in an online survey, where they got to choose their preferred hat, top hat or beret. Out of the males that participated, twelve of them prefer a beret. Fifteen of the females chose top hat as their preference.

1

Determine the categories

First, determine the two categories of the table and draw it without frequencies. Here, the participants gave their hat preference and their gender, which are then the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male. This gives the following table.

Hat preference
Top hat Beret Total
Gender Male
Female
Total

The "total" row and columns are included to make room for the marginal frequencies.


2

Fill the table with given data


The given joint and marginal frequencies can now be added to the table.

Hat preference
Top hat Beret Total
Gender Male
Female
Total


3

Find any missing frequencies


Using the given frequencies, more information can potentially be found by reasoning. For instance, out of the males prefers berets, which means that males prefer top hats. Thus, there are males and females who prefer top hats, making a total of participants that prefer top hats. Continuing this reasoning, the entire table can be completed.

Hat preference
Top hat Beret Total
Gender Male
Female
Total


Concept

Joint and Marginal Relative Frequencies

A joint relative frequency is the ratio of a joint frequency and the total number of values or observations. Similarly, a marginal relative frequency is the ratio of a marginal frequency and the total. For the example above, the joint and marginal relative frequencies are found by dividing the frequencies by the number of participants.

Hat preference
Top hat Beret Total
Gender Male
Female
Total
Notice that, ignoring the error margin introduced by rounding, a marginal relative frequency can be found by adding a row or column of joint marginal frequencies. This table shows, for instance, that female's with a preference for berets make up about of the participants.
Concept

Conditional Relative Frequency

A conditional relative frequency is the ratio of a joint frequency and either of its corresponding two marginal frequencies. Alternatively, it can be calculated using relative joint and marginal frequencies. As an example, the following data will be used.

Driver's license
Yes No Total
Car Yes
No
Total

Using the column totals, the left column of joint frequencies should be divided by and the right column by Since the column totals are used, the sum of the conditional relative frequencies will be

Driver's license
Yes No
Car Yes
No
This table shows that, for instance, out of all the participants with a driver's license, about of them own a car, and out of those without a driver's license, do not have a car.
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Exercise

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
No
Total
Show Solution
Solution

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, and that are used as the denominators.

Vegetarian
Yes No
Pet Yes
No
Total

Note that the sum of the conditional relative frequencies in each column is equal to 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
No
Total

In that column, said "Yes" to having a pet and said "no". Thus, the probability that a vegetarian has a pet is

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