AI Tutor
Imagine rolling a standard six-sided die under a cup. If a friend peeks and tells you the result is an even number, what are the chances you rolled a six?
Before the hint, your chance was 16\frac{1}{6}. But with the new information, the possible outcomes shrink from {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} to just {2,4,6}\{2, 4, 6\}, making the probability 13\frac{1}{3}.

The Restricted Universe

In probability, conditional probability measures the likelihood of an event AA occurring given that another event BB has already happened, written as P(A∣B)P(A|B).
This core mechanism is known as sample space restriction: conditioning on an event BB shrinks or restricts the universal sample space SS down strictly to the subset where BB occurs, making P(B)P(B) the new denominator.
This geometric relationship gives us the formal conditional probability formula:
P(A∣B)=P(A∩B)P(B),where P(B)>0P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{where } P(B) > 0
In the 18th century, English statistician Thomas Bayes and French mathematician Pierre-Simon Laplace pioneered this relationship to determine how new evidence should rationally update existing odds.
How do we compute conditional probability when events are given as abstract set values?

Worked Example: Set Probabilities

Suppose P(A)=0.7P(A) = 0.7, P(B)=0.5P(B) = 0.5, and the probability of at least one event occurring is P(A∪B)=0.85P(A \cup B) = 0.85. Let's find P(A∣B)P(A|B).
Before applying the conditional formula, we need the intersection P(A∩B)P(A \cap B). We can find it using the addition rule of probability: