Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred.

Definition

The conditional probability of given (denoted as ) is defined as:

: The probability that both and occur.

: The probability that occurs.

This definition assumes that because dividing by zero is undefined.

Intuition Behind Conditional Probability

Conditional probability modifies the perspective by reducing the sample space to only those outcomes where occurs. Essentially, it tells us how likely is under the condition that has occurred.

Suppose we roll a fair six-sided die.

Let be the event the number is even ().

Let be the event the number is greater than 3 ().

The sample space is .

The intersection of and is .

The probability of is .

The conditional probability of given is:

This means that if we know the number rolled is greater than 3, there is a chance it is also even.

Key Properties of Conditional Probability

Non-Negativity: .

Normalization: If is known to occur, .

Multiplication Rule: Rearranging the formula for conditional probability gives:

This is useful when and are known, and needs to be calculated.

Example 1: Medical Testing

Imagine a medical test for a disease with the following probabilities:

Probability a person has the disease: .

Probability the test is positive if the person has the disease (sensitivity): .

Probability the test is positive if the person does not have the disease (false positive rate): .

What is the probability a person has the disease given a positive test result ()?

Using Bayes’ theorem, which relies on conditional probability:

We calculate using the law of total probability:

Substitute the values:

Now, calculate :

Thus, even with a positive test result, the probability of actually having the disease is only 16.1%.

Example 2: Card Drawing

A card is drawn from a standard 52-card deck.

Let be the event the card is a King ().

Let be the event the card is a face card ().

The probability of drawing a face card is

The probability of drawing a King and a face card () is

The conditional probability of given is:

This means that if a face card is drawn, there is a chance it is a King.