Conditional Probability

Definition

The conditional probability of event $A$ given event $B$ has occurred is the probability that $A$ happens within the restricted situation where $B$ is already known to have happened.

It is written as:

$$P(A \mid B)$$

and defined by the formula:

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{provided } P(B) > 0$$

Where:

• $P(A \mid B)$ = probability of $A$ given $B$
• $P(A \cap B)$ = probability that both $A$ and $B$ occur together
• $P(B)$ = probability that event $B$ occurs

This formula means that once event $B$ has occurred, the new sample space becomes only the outcomes inside $B$. Then we ask: among those outcomes, how many also belong to $A$?

Main Content

1. Conditional Probability Formula

• The most important idea in conditional probability is that the denominator changes after new information is known.
• Instead of using the full sample space, we only consider outcomes that satisfy the condition $B$.

Formula interpretation

$$P(A \mid B) = \frac{\text{outcomes in both } A \text{ and } B}{\text{outcomes in } B}$$

This is not the same as $P(A)$. The probability of $A$ may increase, decrease, or stay the same after knowing $B$.

Example

Suppose a class has 30 students:

• 12 are girls
• 8 students like mathematics
• 5 are girls who like mathematics

Let:

• $A$ = student likes mathematics
• $B$ = student is a girl

Then:

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{5/30}{12/30} = \frac{5}{12}$$

So, if we already know the student is a girl, the probability that she likes mathematics is $5/12$.

Key idea

• Conditional probability is not about all outcomes.
• It is about restricted outcomes after given information.

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2. Intersection and Multiplication Rule

• Conditional probability is closely connected to the probability that two events occur together.
• Rearranging the definition gives the multiplication rule:

$$P(A \cap B) = P(B)\cdot P(A \mid B)$$

Also:

$$P(A \cap B) = P(A)\cdot P(B \mid A)$$

This formula is extremely useful when the probability of a combined event is easier to find using conditional probability than by direct counting.

Example

A bag contains 4 red balls and 6 blue balls. Two balls are drawn without replacement.

Let:

• $A$ = first ball is red
• $B$ = second ball is red

To find $P(A \cap B)$:

• $P(A)=4/10$
• After one red ball is drawn, 3 red balls remain out of 9 balls, so $P(B \mid A)=3/9$

Thus:

$$P(A \cap B)=P(A)\cdot P(B \mid A)=\frac{4}{10}\cdot\frac{3}{9}=\frac{12}{90}=\frac{2}{15}$$

Important insight

• The multiplication rule helps when events are dependent, meaning one event affects the other.
• It is a foundation for probability trees and sequential events.

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3. Independence and Dependence

• Two events are independent if the occurrence of one does not affect the probability of the other.
• In that case:

$$P(A \mid B) = P(A)$$

and also:

$$P(A \cap B) = P(A)P(B)$$

If the probability changes after knowing another event occurred, then the events are dependent.

Example of independence

Toss a fair coin and roll a die.

• $A$ = getting heads
• $B$ = getting a 6

Since the coin toss does not affect the die roll:

$$P(A \mid B)=P(A)=\frac{1}{2}$$

Example of dependence

If a card is drawn from a deck without replacement:

• $A$ = first card is an ace
• $B$ = second card is an ace

Knowing that the first card is an ace changes the chance that the second card is an ace, so the events are dependent.

Why this matters

• Independence is often tested using conditional probability.
• If $P(A \mid B) \ne P(A)$, the events are not independent.

Working / Process

1. Identify the events clearly

• Decide which event is the condition and which event is the result.
• Write them as $A$ and $B$ or use meaningful labels.

2. Find the restricted sample space

• Focus only on outcomes where the condition event $B$ occurs.
• Count the total outcomes in $B$ and the outcomes in both $A$ and $B$.

3. Apply the conditional probability formula

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

• Simplify carefully.
• If necessary, use a tree diagram or table to organize the outcomes.

Example using a tree-style reasoning

A jar contains 3 green and 2 yellow marbles. Two marbles are drawn without replacement.

First draw:
G (3/5)
  ├─ Second draw G (2/4)
  └─ Second draw Y (2/4)

Y (2/5)
  ├─ Second draw G (3/4)
  └─ Second draw Y (1/4)

If we want the probability of drawing a yellow marble on the second draw given that the first draw was green:

$$P(Y_2 \mid G_1)=\frac{2}{4}=\frac{1}{2}$$

This is found by looking only at the branch where the first draw is green.

Advantages / Applications

• Helps make better predictions when new information is available
• Used in medical testing, where the chance of disease changes after a positive result
• Important in machine learning, artificial intelligence, and data analysis for updating probabilities
• Used in quality control to estimate defect rates under certain conditions
• Applied in risk assessment, reliability engineering, genetics, finance, and weather forecasting

Real-world examples

Medical testing:
If a test result is positive, conditional probability helps determine the chance the patient truly has the disease.

Weather forecasting:
The probability of rain given high humidity may be much higher than the general probability of rain.

Manufacturing:
If a product comes from a specific machine, the probability of defect may depend on that machine.

Card and dice problems:
Conditional probability simplifies sequential event calculations.

Summary

• Conditional probability measures the chance of an event when another event is known to have happened.
• It is calculated using $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$.
• It is closely related to dependent events and the multiplication rule.
• Important terms to remember: conditional probability, intersection, dependence, independence, sample space