Conditional Probability
So far, we have calculated probabilities assuming no additional information. In many real-life situations, however, we already know something has happened.
Conditional probability helps us calculate the probability of an event occurring given that another event has already occurred.
What Is Conditional Probability?
Conditional probability is the probability of event A occurring, given that event B has already occurred.
It is written as:
P(A | B)
This is read as “the probability of A given B.”
Why Conditional Probability Matters
Once some information is known, the sample space changes. That means probabilities must be recalculated using only the remaining possible outcomes.
This concept is widely used in:
- Medical testing
- Risk analysis
- Machine learning
- Decision-making under uncertainty
Basic Conditional Probability Formula
The formula for conditional probability is:
P(A | B) = P(A and B) ÷ P(B)
This formula applies when P(B) is not zero.
Numerical Example (Cards)
A card is drawn at random from a standard deck of 52 cards.
Suppose we know that the card drawn is a heart.
What is the probability that the card is a queen?
There are:
- 13 hearts in total
- Only 1 queen of hearts
So, the probability is:
P(Queen | Heart) = 1 ÷ 13
Notice how the total outcomes reduced from 52 to 13 because we already know the card is a heart.
Numerical Example (Using the Formula)
Let:
- P(A and B) = 0.2
- P(B) = 0.5
Then:
P(A | B) = 0.2 ÷ 0.5 = 0.4
This means that once B has occurred, the chance of A occurring is 40%.
Real-World Example (Medical Testing)
Consider a medical test for a disease.
- Event A: Person has the disease
- Event B: Test result is positive
The probability that a person actually has the disease given that the test is positive is a conditional probability.
This concept is extremely important because a positive test does not always mean the person has the disease.
Conditional Probability vs Independent Events
If two events are independent, knowing that one event occurred does not change the probability of the other.
In that case:
P(A | B) = P(A)
If the probability changes after knowing B, then the events are dependent.
Quick Check
If P(A | B) = P(A), what does it say about events A and B?
Practice Quiz
Question 1:
What does P(A | B) represent?
Question 2:
If P(A and B) = 0.15 and P(B) = 0.3, what is P(A | B)?
Question 3:
Does conditional probability always mean events are dependent?
Mini Practice
A bag contains 5 red balls and 5 blue balls. Two balls are drawn without replacement.
- What is the probability the second ball is red, given the first was red?
What’s Next
In the next lesson, we will study Random Variables, which help us move from probability concepts to mathematical models.