Conditional Probability
Conditional probability is the probability of an event B occurring,
given that another event A has already occurred. It is written as:
Example 1: Drawing Cards Without Replacement
Let’s say we draw two cards from a standard deck (52 cards) without
replacement. When we draw cards without replacement, it means that once a
card is drawn, it is not put back into the deck before drawing the next card. This
affects the probability of future draws because the total number of cards in the
deck decreases.
Question: What is the probability of drawing an Ace on the second draw,
given that the first card drawn was an Ace?
Step-by-Step Calculation:
 Total number of cards = 52
 Total Aces in the deck = 4
 Event A: Drawing an Ace on the first draw :

Event B: Drawing an Ace on the second draw given that A happened :
Since we already removed one Ace from the deck, we now have:

3 Aces left

51 cards remaining
Example 2: Probability of a Student Passing a Course Given They Studied
Scenario: A teacher observes that 80% of students study for exams. Among
those who study, 90% pass the course. Among those who do not study, only
30% pass.
Question: If a student studied, what is the probability that they pass the
course?
Let:

A = Student studies → P(A)=0.8P(A) = 0.8P(A)=0.8

B = Student passes the course

90% of the student who study for exam get pass. So
P(A∩B) = 90 % of student who studied for exam
of student who studied for exam = 80%
P(A∩B) = 90/100 * 80/100
P(A∩B) = 0.9 * 0.8
P(B∣A)= (0.9 * 0.8)
0.8
P(A∩B) = 0.9
Thus, if we know a student studied, the conditional probability of them passing
is 90% (or 0.9).
Example 3: You roll a six-sided dice. What is the probability that the number
is even, given that it is greater than 2?
Solution:
 Sample space: S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}S={1,2,3,4,5,6}
 Let A = Number is greater than 2 → A={3,4,5,6}A = \{3, 4, 5,
6\}A={3,4,5,6}
 Let B = Number is even → B={2,4,6}B = \{2, 4, 6\}B={2,4,6}
Now, given that the number is greater than 2, the new sample space is A = {3,
4, 5, 6}.
 The even numbers in this new space are {4, 6}.
So, conditional probability:
3. Drawing Balls from a Bag
Scenario:
A bag contains 5 red and 3 blue balls. You draw one ball, then another,
without replacement.
Question: What is the probability that the second ball is red, given that the
first ball was red?
Solution:

Total balls = 8

Probability of drawing red first:

After drawing one red ball, only 4 red remain, and total balls are 7.

Probability of drawing a red on the second draw given the first was red:
So, the conditional probability is 4/7.
Difference Between Conditional Probability (P(B∣A)) and Dependent
Probability (P(A∩B))
Example: Drawing Cards Without Replacement

Conditional Probability:
o

What is the probability of drawing an Ace on the second draw
given that the first card was an Ace?
Dependent Probability:
o What is the probability of drawing two Aces in a row?