Conditional Probability

You do it all the time without realizing it. Your friend texts that they are stuck in traffic, and suddenly you update your mental estimate of when they will arrive. The doctor says your test came back positive, and you start wondering what that really means for your health. Weather radar shows a storm heading your direction, and you adjust your plans for the afternoon.

This is conditional probability in action: updating what you believe based on new information. Probability is not static. It changes as you learn more about the world. A 10% chance of rain becomes 90% once you see dark clouds rolling in. The odds that your favorite team wins change dramatically once the star player gets injured.

Conditional probability formalizes this intuitive process. It answers questions like: “Given that I know X happened, what is the probability of Y?” This simple shift - from asking about probability in general to asking about probability given specific information - turns out to be one of the most powerful and practical ideas in all of mathematics.

Core Concepts

What Is Conditional Probability?

Conditional probability is the probability of an event occurring given that another event has already occurred (or is known to be true). It answers the question: “Now that I know this, what are the chances of that?”

Think about drawing cards from a deck. Before any cards are drawn, the probability of drawing an Ace is $\frac{4}{52}$. But suppose someone tells you that the card drawn is a face card (Jack, Queen, or King). Now you know the Ace is impossible because Aces are not face cards. The probability of having drawn an Ace, given that you drew a face card, is 0.

Or consider this: you flip two coins and someone tells you that at least one came up heads. What is the probability that both are heads? Without this information, the probability of two heads is $\frac{1}{4}$. But with the information that at least one is heads, the probability changes to $\frac{1}{3}$. (We will work through why shortly.)

Conditional probability is about updating your assessment when your available information changes. The original probability is what you believe before learning anything specific. The conditional probability is what you should believe after learning the new information.

Notation: $P(A|B)$

We write conditional probability as $P(A|B)$, which reads as “the probability of A given B.”

$$P(A|B) = \text{probability of } A \text{ occurring, given that } B \text{ has occurred}$$

The vertical bar “|” means “given that” or “conditional on.” Everything to the right of the bar is what you know or assume to be true. Everything to the left is what you want to find the probability of.

Some examples:

• $P(\text{rain tomorrow} | \text{cloudy today})$ = probability it rains tomorrow given that today is cloudy
• $P(\text{pass exam} | \text{studied 10+ hours})$ = probability of passing given that you studied at least 10 hours
• $P(\text{disease} | \text{positive test})$ = probability of having the disease given that your test was positive

Reading conditional probability notation correctly is crucial. Pay close attention to what is on each side of the bar:

• The event on the left is what you are finding the probability of
• The event on the right is what you already know or are assuming

The Conditional Probability Formula

Here is the formal definition of conditional probability:

$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$$

This formula makes intuitive sense. When you know that B has happened, you are restricting your attention to only those outcomes where B occurred. The sample space has shrunk from “everything that could happen” to “only things where B happened.” Within this restricted space, you want to know how often A also happens.

Let us break this down:

• $P(A \text{ and } B)$ is the probability that both A and B occur
• $P(B)$ is the probability that B occurs
• The ratio tells you what fraction of the “B outcomes” are also “A outcomes”

Important: This formula only works when $P(B) > 0$. You cannot condition on an impossible event. (What would it mean to ask the probability of something given that an impossible thing happened?)

Here is a simple example. Suppose you roll a fair die. Let A be “roll a 6” and B be “roll an even number.” What is $P(A|B)$?

We need:

• $P(A \text{ and } B)$: probability of rolling a 6 that is also even. Since 6 is even, this is just $P(\text{roll a 6}) = \frac{1}{6}$.
• $P(B)$: probability of rolling an even number (2, 4, or 6) = $\frac{3}{6} = \frac{1}{2}$.

Therefore: $$P(A|B) = \frac{P(A \text{ and } B)}{P(B)} = \frac{1/6}{1/2} = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3}$$

This makes sense. If you know you rolled an even number, then you know you got 2, 4, or 6. One out of these three equally likely possibilities is 6, so the conditional probability is $\frac{1}{3}$.

Two-Way Tables and Conditional Probability

Two-way tables (also called contingency tables) organize data by two categories and are perfect for calculating conditional probabilities. They show you exactly how many items fall into each combination of categories.

Here is an example. Suppose a school surveyed 200 students about their study habits and test performance:

	Passed Test	Failed Test	Total
Studied	80	20	100
Did Not Study	30	70	100
Total	110	90	200

From this table, you can calculate many conditional probabilities:

$P(\text{passed} | \text{studied})$: Among students who studied, what fraction passed?

• Look at the “Studied” row: 80 passed out of 100 who studied. $$P(\text{passed} | \text{studied}) = \frac{80}{100} = 0.80 = 80%$$

$P(\text{passed} | \text{did not study})$: Among students who did not study, what fraction passed?

• Look at the “Did Not Study” row: 30 passed out of 100 who did not study. $$P(\text{passed} | \text{did not study}) = \frac{30}{100} = 0.30 = 30%$$

Notice the huge difference. Studying more than doubled the probability of passing. This is exactly the kind of insight conditional probability reveals.

$P(\text{studied} | \text{passed})$: Among students who passed, what fraction had studied?

• Look at the “Passed Test” column: 80 studied out of 110 who passed. $$P(\text{studied} | \text{passed}) = \frac{80}{110} = \frac{8}{11} \approx 0.727 = 72.7%$$

The key technique with two-way tables: the condition (what comes after the “|”) tells you which row or column to restrict your attention to. Then you find the relevant count within that row or column divided by the row or column total.

Tree Diagrams for Conditional Probability

Tree diagrams visualize sequential events and conditional probabilities beautifully. Each branch represents a possible outcome, and the probability on each branch is the conditional probability given the path you have taken so far.

Consider this scenario: a bag contains 3 red marbles and 2 blue marbles. You draw two marbles without replacement (you do not put the first marble back). What are the probabilities of various outcomes?

Here is how to build the tree:

First draw:

• $P(\text{Red}_1) = \frac{3}{5}$ (3 red out of 5 total)
• $P(\text{Blue}_1) = \frac{2}{5}$ (2 blue out of 5 total)

Second draw, given first was Red:

• Now there are 2 red and 2 blue left (4 total)
• $P(\text{Red}_2 | \text{Red}_1) = \frac{2}{4} = \frac{1}{2}$
• $P(\text{Blue}_2 | \text{Red}_1) = \frac{2}{4} = \frac{1}{2}$

Second draw, given first was Blue:

• Now there are 3 red and 1 blue left (4 total)
• $P(\text{Red}_2 | \text{Blue}_1) = \frac{3}{4}$
• $P(\text{Blue}_2 | \text{Blue}_1) = \frac{1}{4}$

To find the probability of any complete path, multiply the probabilities along the branches:

• $P(\text{Red, then Red}) = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$
• $P(\text{Red, then Blue}) = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$
• $P(\text{Blue, then Red}) = \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}$
• $P(\text{Blue, then Blue}) = \frac{2}{5} \times \frac{1}{4} = \frac{2}{20} = \frac{1}{10}$

Notice these probabilities sum to 1: $\frac{3}{10} + \frac{3}{10} + \frac{3}{10} + \frac{1}{10} = \frac{10}{10} = 1$. This is a good check that your calculations are correct.

Testing for Independence Using Conditional Probability

Two events A and B are independent if learning that one occurred does not change the probability of the other. Mathematically:

$$\text{A and B are independent if and only if } P(A|B) = P(A)$$

Equivalently: $$\text{A and B are independent if and only if } P(B|A) = P(B)$$

When events are independent, conditional probability equals unconditional probability - the information about one event is useless for predicting the other.

Here is how to test for independence:

1. Calculate $P(A)$ (unconditional probability of A)
2. Calculate $P(A|B)$ (conditional probability of A given B)
3. If $P(A) = P(A|B)$, the events are independent. If not, they are dependent.

Returning to our study habits example:

• $P(\text{passed}) = \frac{110}{200} = 0.55 = 55%$ (overall pass rate)
• $P(\text{passed} | \text{studied}) = \frac{80}{100} = 0.80 = 80%$

Since $0.80 \neq 0.55$, studying and passing are not independent. Knowing someone studied increases your estimate of their probability of passing. This confirms what common sense tells us: studying helps.

For independent events, another useful formula emerges. If A and B are independent: $$P(A \text{ and } B) = P(A) \times P(B)$$

This is the multiplication rule for independent events. Dependent events require the more general rule: $$P(A \text{ and } B) = P(A) \times P(B|A) = P(B) \times P(A|B)$$

Bayes’ Theorem Introduction

Bayes’ Theorem is a formula for “reversing” conditional probability. If you know $P(B|A)$, it helps you find $P(A|B)$. This is incredibly useful because sometimes one conditional probability is much easier to determine than the other.

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Let us understand why you might need this. Consider medical testing:

• Doctors know $P(\text{positive test} | \text{have disease})$ - this is the test’s sensitivity
• But patients want to know $P(\text{have disease} | \text{positive test})$ - what does a positive result mean?

These are not the same thing. Bayes’ Theorem lets you go from one to the other.

The terms in Bayes’ Theorem have names:

• $P(A)$ is the prior probability - your probability assessment before the new information
• $P(A|B)$ is the posterior probability - your updated probability after learning B
• $P(B|A)$ is the likelihood - how likely is the evidence B if A were true
• $P(B)$ is the marginal probability of B - the total probability of observing B

Often, $P(B)$ must be calculated using the law of total probability: $$P(B) = P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c)$$

This accounts for all the ways B can occur: either A happens and B follows, or A does not happen and B follows anyway.

Common Confusion: $P(A|B)$ vs. $P(B|A)$

One of the most common errors in probability is confusing $P(A|B)$ with $P(B|A)$. These are almost never equal, and mixing them up can lead to seriously wrong conclusions.

Here is a vivid example:

• $P(\text{wet street} | \text{rained})$ is very high - if it rained, the street is almost certainly wet
• $P(\text{rained} | \text{wet street})$ is lower - the street could be wet from a sprinkler or street cleaner

The direction matters enormously.

This confusion shows up in real-world reasoning errors:

• Prosecutors sometimes argue that since $P(\text{evidence} | \text{guilty})$ is high, $P(\text{guilty} | \text{evidence})$ must also be high. This is the “prosecutor’s fallacy.”
• Medical patients learn that $P(\text{positive} | \text{no disease})$ is low (false positive rate is small), and wrongly assume $P(\text{no disease} | \text{positive})$ is also low.

Always be clear about what you are conditioning on and what you are trying to find. If you need to reverse a conditional probability, use Bayes’ Theorem.

Notation and Terminology

Term	Meaning	Example
$P(A | B)$	Probability of A given that B occurred	Restricted sample space
Conditional	Depending on known information	Probability changes with knowledge
Two-way table	Table organizing data by two categories	Rows and columns with counts
Bayes’ Theorem	$P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}$	Updating probability with evidence
Prior probability	Probability before new information	$P(\text{disease})$ before testing
Posterior probability	Probability after new information	$P(\text{disease} | \text{positive})$
Independent events	Events where one does not affect the other	$P(A | B) = P(A)$
Dependent events	Events where one affects the other	Studying affects passing probability

Examples

Key Properties and Rules

The Conditional Probability Formula

$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}, \quad \text{provided } P(B) > 0$$

Rearranging gives the general multiplication rule: $$P(A \text{ and } B) = P(B) \cdot P(A|B) = P(A) \cdot P(B|A)$$

Independence Test

Events A and B are independent if any of these equivalent statements hold:

• $P(A|B) = P(A)$
• $P(B|A) = P(B)$
• $P(A \text{ and } B) = P(A) \cdot P(B)$

Bayes’ Theorem

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

Where $P(B)$ can be expanded as: $$P(B) = P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c)$$

Properties of Conditional Probability

Conditional probabilities follow the same rules as regular probabilities:

• $0 \leq P(A|B) \leq 1$
• $P(B|B) = 1$ (if B happened, then B happened)
• $P(A^c|B) = 1 - P(A|B)$ (conditional complement rule)

Common Mistakes to Avoid

1. Confusing $P(A|B)$ with $P(B|A)$ - these are usually different
2. Assuming independence - always check; do not assume events are independent
3. Forgetting to restrict the sample space - in conditional probability, your “universe” shrinks to only outcomes where the condition holds
4. Multiplying probabilities incorrectly - use $P(A) \cdot P(B)$ only for independent events; otherwise use $P(A) \cdot P(B|A)$

Real-World Applications

Medical Diagnosis

Medical testing is a prime example of conditional probability. When you get a positive test result, what you really want to know is the probability you have the condition given the positive test. But test accuracy is typically reported the other way: the probability of a positive test given you have the condition.

This is exactly where Bayes’ Theorem shines. It helps doctors and patients interpret test results correctly, taking into account:

• How accurate the test is (sensitivity and specificity)
• How common the disease is in the population (prevalence)

For rare diseases, even accurate tests produce mostly false positives. For common conditions, even moderate tests are informative. Understanding conditional probability helps make sense of medical information.

Spam Filters

Email spam filters use conditional probability extensively. The filter learns probabilities like:

• $P(\text{contains “free money”} | \text{spam})$ - very high
• $P(\text{contains “free money”} | \text{not spam})$ - very low

When a new email arrives, the filter uses Bayes’ Theorem to calculate $P(\text{spam} | \text{contains “free money”})$ and similar probabilities for other words and features. If the overall probability of spam given all the features is high enough, the email goes to your spam folder.

This approach, called Naive Bayes classification, is one of the most successful applications of probability theory to real-world problems.

Weather Prediction

Weather forecasting relies heavily on conditional probability. Meteorologists calculate probabilities like:

• $P(\text{rain tomorrow} | \text{low pressure system approaching})$
• $P(\text{severe weather} | \text{certain atmospheric conditions})$

Modern forecasting combines historical data (experimental probability from similar past conditions) with physical models to produce conditional probability estimates. When you see “40% chance of rain,” that is a conditional probability given current atmospheric conditions.

Legal Reasoning

In courtrooms, evidence must be evaluated carefully using conditional probability. Lawyers and judges deal with questions like:

• $P(\text{evidence} | \text{guilty})$ vs. $P(\text{guilty} | \text{evidence})$

The “prosecutor’s fallacy” occurs when these are confused. Just because evidence would be very likely if the defendant were guilty does not mean the defendant is likely guilty given the evidence. The prior probability matters enormously.

DNA evidence is a famous example. A DNA match might have probability 1 in a million if the person is innocent. But this does not mean there is a 1 in a million chance of innocence, because we must also consider how likely it was that this particular person committed the crime before we saw the DNA evidence.

Self-Test Problems

Summary

• Conditional probability $P(A|B)$ measures the probability of A given that B has occurred. It answers: “Now that I know B happened, what are the chances of A?”

• The conditional probability formula is $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$. The condition restricts your sample space to only outcomes where B occurred.

• Two-way tables make conditional probability calculations straightforward: the condition tells you which row or column to focus on.

• Tree diagrams visualize sequential conditional probabilities. Multiply along branches to find path probabilities; add paths to find total probabilities.

• Events are independent if $P(A|B) = P(A)$ - learning B happened does not change the probability of A. Test this by comparing conditional and unconditional probabilities.

• Bayes’ Theorem reverses conditional probabilities: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$. It is essential when you know one conditional probability but need the reverse.

• Prior and posterior probability: The prior is your belief before new information; the posterior is your updated belief after learning new evidence.

• $P(A|B)$ and $P(B|A)$ are different. Confusing them is a common and serious error. Always be clear about what you are conditioning on versus what you are finding the probability of.

• Real-world applications include medical testing, spam filtering, weather prediction, and legal reasoning - anywhere that new information should update probability assessments.