2.3 Calculate marginal and conditional probabilities | Athabasca University MATH 215: Introduction to Statistics

Athabasca University MATH 215: Introduction to Statistics

Probability

MATH 215, Study Topic 2.3

A probability describes how likely an event is. It is a number from 0 to 1: 0 means the event cannot occur, and 1 means it is certain. In this lesson, counts in a two-way table are used to calculate marginal and conditional probabilities. A marginal probability describes an event in the whole group. A conditional probability describes an event within a group restricted by a condition. The key choice in either calculation is the denominator: use the overall total for a marginal probability and the total for the stated condition for a conditional probability.

What you will learn

Distinguish marginal probabilities from conditional probabilities.
Calculate marginal probabilities from totals in a two-way table.
Calculate conditional probabilities using the total for the stated condition.
Interpret each probability in the group and context it describes.

1. Read the table and identify events

A two-way table organizes counts according to two categorical variables. A categorical variable sorts people or items into named groups, such as morning or evening, or yes or no. Each inside cell counts people or items that belong to both its row category and its column category. Row and column totals summarize one variable across the categories of the other variable. These are called marginal totals.

An event is an outcome or group whose probability we want to find. For example, selecting a member who attends in the morning is an event. For a table of counts, imagine choosing one person at random from the group represented by the table. The population is the full group we want to understand; a sample is the part of it that was observed. If the table describes a sample, its proportions directly describe that sample, but should not automatically be treated as exact probabilities for a wider population.

Before calculating, check that the table represents the group named in the question, its categories are clear, and each person or item is counted once. Counts must be nonnegative, and the row and column totals must agree with the overall total. A conditional probability also requires a nonempty given group. If that group's total is zero, division by zero is not possible and the conditional probability is undefined.

An inside cell counts items in both its row and column categories.
A marginal total summarizes one category across the other variable.
The overall total is the number of items represented by the table.

2. Calculate a marginal probability

A marginal probability is the probability of an event in the whole group, without restricting attention to another category. For a table of counts, divide the event's row or column total by the overall total. The event total is appropriate because it includes everyone in the event, regardless of their category on the other variable.

For example, the event "attends in the morning" includes morning attendees who answered yes and morning attendees who answered no. Its count is the morning row total. The denominator is everyone represented in the table. In general, for event

A

, divide the count in

A

by the overall total. The notation

P(A)

means the probability of event

A

.

A count is not a measurement such as metres or kilograms. Dividing one count by another gives a proportion, which can be reported as a decimal or percentage. The basic algebra is ordinary division: a fraction means its numerator divided by its denominator, so

3/4=0.75

. Check that a calculated probability lies from 0 to 1 inclusive. If categories divide the whole group into all possible outcomes, their marginal probabilities add to 1. Keep several decimal places during the calculation and round only the final answer.

For an event

A

, the marginal probability is the count in

A

divided by the overall total.

P(A)=\frac{n_A}{n}

Use the event's row or column total as the numerator.
Use the overall total as the denominator when there is no condition.
Check that the result is between 0 and 1.

3. Calculate a conditional probability

A conditional probability is the probability of an event within a group specified by a condition. In the notation

P(A\mid B)

, the vertical bar is read as "given." Thus,

P(A\mid B)

means the probability of event

A

given event

B

.

For a table of counts, the condition

B

identifies the restricted group and therefore the denominator. The numerator counts members of that group who also meet event

A

. In words, divide the count in both

A

and

B

by the count in

B

. This calculation requires the given group to have a positive total. It uses counts directly, so there is no need to calculate a separate probability first.

The order of the events matters. "Yes given morning" asks about yes responses among morning attendees. "Morning given yes" asks about morning attendees among people who answered yes. The overlap count may be the same, but the denominators differ, so the answers need not match. Read the wording carefully before selecting a total.

For events

A

and

B

, the conditional probability is the count in both events divided by the count in the given event

B

.

P(A\mid B)=\frac{n_{A\cap B}}{n_B}

The event after "given" identifies the restricted group and denominator.
The numerator counts items in both the target event and the given group.
Reversing the condition changes the question and usually the answer.

4. Choose the denominator and interpret the result

Start by translating the question into an event and checking whether it includes a condition such as "among," "given," or "of those who." With no condition, use the overall total. With a condition, use the total for the group named after "given." Choose the denominator before dividing; this helps prevent using a cell count when the question calls for a row, column, or overall total.

The conditions for these calculations are straightforward: the counts must refer to the group in the question, the table categories and totals must be consistent, and the denominator must be positive. The algebra is ordinary division, so do not divide by zero. Keep four to six decimal places in intermediate work and round only the final answer. Then state what the result means in context. A conditional probability should identify both the event and the group being considered, not just give a decimal.

Translate the wording into an event and, if present, a given group.
Choose the denominator before calculating.
Check the result and interpret it in context.

Club members by meeting time and survey response

Meeting time	Yes	No	Total
Morning	30	18	48
Evening	20	12	32
Total	50	30	80

Worked example

Club attendance and survey responses

A club records 80 members by usual meeting time and whether they answered yes to a survey question. Use the table to find (1) the marginal probability that a randomly selected member attends in the morning, (2) the probability of a yes response given morning attendance, and (3) the probability of morning attendance given a yes response.

Check the counts
Each member is counted once in one meeting-time and response combination. The row and column totals agree with the overall total. The groups used as conditions have positive totals, so the requested conditional probabilities are defined.
$30+18=48, 20+12=32, 48+32=80$
Calculate morning attendance
Morning attendance is asked about without a condition, so use the morning total over all 80 members. The result is the proportion of the full club represented by morning attendees.
$P(\text{morning})=\frac{48}{80}=0.600000$
Calculate yes given morning
"Given morning" restricts attention to the 48 morning attendees. Of those, 30 answered yes. Divide the overlap count by the morning total.
$P(\text{yes}\mid\text{morning})=\frac{30}{48}=0.625000$
Calculate morning given yes
Now the condition is a yes response. There are 50 yes responses in total, and 30 of those members attend in the morning. Use the yes total as the denominator. Round only the final answer.
$P(\text{morning}\mid\text{yes})=\frac{30}{50}=0.600000$

Answer: The marginal probability of morning attendance is 0.600, or 60.0%. Among morning attendees, the probability of a yes response is 0.625, or 62.5%. Among members who answered yes, the probability of morning attendance is 0.600, or 60.0%.

Check: The conditional probabilities answer different questions: yes given morning uses a denominator of 48, while morning given yes uses a denominator of 50. Each result is between 0 and 1.

Common mistakes and how to avoid them

Using the overall total as the denominator for a conditional probability.

Correction: Use the total for the group named after "given." For yes given morning, the denominator is the morning total.

Reversing the order of the events in a conditional probability.

Correction: The event after "given" determines the denominator group. Read the wording before choosing the total.

Using a cell count as the denominator without checking the question.

Correction: A cell count often supplies the overlap in the numerator. The denominator is the relevant event total, condition total, or overall total.

Reporting a decimal without explaining what it describes.

Correction: Name the event and, for a conditional probability, the group within which the probability applies.

Lesson summary

A marginal probability is an event total divided by the overall total.
A conditional probability is an overlap count divided by the total for the stated condition.
The word "given" identifies the restricted group and denominator.
Check the table totals, make sure a conditional denominator is positive, and interpret the result in context.

Check your understanding

Question 1

In the club table, what is the marginal probability that a randomly selected member answered yes?

0.375
0.600
0.625
0.789

Show answer and explanation

0.625

Yes is a marginal event. The yes total is 50 out of 80 members, so the probability is

50/80=0.625

.

Question 2

Among members who answered no, what is the probability that a member attends in the evening?

0.150
0.400
0.600
0.750

Show answer and explanation

0.400

The given group is the 30 members who answered no. Of them, 12 attend in the evening, so the probability is

12/30=0.400

.

Key terms

Event: A stated outcome or group whose probability is being considered.
Two-way table: A table that organizes counts according to categories of two variables.
Marginal total: A row or column total that summarizes counts across the other variable.
Marginal probability: The probability of an event in the whole group, without a condition.
Conditional probability: The probability of an event within a group specified by a condition.

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2.3 · Calculate marginal and conditional probabilities