Conditional Probability Questions
Question 1: A bag contains 5 red balls and 7 blue balls. Two balls are drawn without replacement. What is the probability that the second ball drawn is red, given that the first ball drawn was red?
Solution:
Let the events be,
Event A: The first ball drawn is red.
Event B: The second ball drawn is red.
P(A) = 5/12
and P(B) = 4/11 (as first ball drawn is already red, thus only 4 red balls remain in the bag)
Therefore, probability of the second ball drawn being red given that the first ball drawn was red is 4/11.
Question 2: A box contains 5 green balls and 3 yellow balls. Two balls are drawn without replacement. What is the probability that both balls are green?
Solution:
Let events be:
Event A: The first ball drawn is green, and
Event B: The second ball drawn is green.
P(A) = 5/8
P(B) = 4/7 (as there are 4 green balls left out of 7)
Thus, probability that both balls drawn are green is (5/8) × (4/7) = 20/56 = 5/14.
Question 3: In a bag, there are 8 red marbles, 4 blue marbles, and 3 green marbles. If one marble is randomly drawn, what is the probability that it is not blue?
Solution:
Let the events be:
Event A: The marble drawn is not blue, and
Event B: The marble drawn is blue.
As A and B are complementary Events, we know
P(A) = 1 – P(B)
⇒ P(A) = 1 – 4/15
⇒ P(A) = (15 – 4)/15
⇒ P(A) = 11/15
Thus, probability of drawing a marble out of bag which is not blue is 11/15.
Question 4: In a survey among a group of students, 70% play football, 60% play basketball, and 40% play both sports. If a student is chosen at random and it is known that the student plays basketball, what is the probability that the student also plays football?
Solution:
Let’s assume there are 100 students in the survey.
Number of students who play football = n(A) = 70
Number of students who play basketball = n(B) = 60
Number of students who play both sports = n(A ∩ B) = 40
To find the probability that a student plays football given that they play basketball, we use the conditional probability formula:
P(A|B) = n(A ∩ B) / n(B)
Substituting the values, we get:
P(A|B) = 40 / 60 = 2/3
Therefore, probability that a randomly chosen student who plays basketball also plays football is 2/3.
Question 5: In a deck of 52 playing cards, 4 cards are drawn without replacement. What is the probability that all 4 cards are aces, given that the first card drawn is an ace?
Solution:
Let the events be,
Event A: The first card drawn is an ace,
Event B: The second card drawn is an ace,
Event C: The third card drawn is an ace, and
Event D: The fourth card drawn is an ace.
P(A) = 4/52 (there are 4 ace out of 52)
P(B | A) = 3/51 (one is already drawn, thus 3 ace left)
P(C | A and B) = 2/50 (two is already drawn, thus 2 ace left)
P(D | A and B and C) = 1/49 (three is already drawn, thus 1 ace left)
To find the probability that all four cards are aces, we multiply the probabilities of the individual events.
P(A and B and C and D) = P(A) × P(B|A) × P(C|A and B) × P(D|A and B and C)
= (4/52) × (3/51) × (2/50) × (1/49)
= 1/270725
Therefore, probability that all 4 cards drawn are aces, given that the first card drawn is an ace, is 1/270725.
Conditional Probability
Conditional probability is one type of probability in which the possibility of an event depends upon the existence of a previous event. As this type of event is very common in real life, conditional probability is often used to determine the probability of such cases.
Conditional probability describes the likelihood of an event (A) happening given that another event (B) has already occurred. In probability notation, this is denoted as A given B, expressed as P(A|B), indicating that the probability of event A is dependent on the occurrence of event B.
To know about conditional probability, we need to be familiar with independent events and dependent events. Let’s understand conditional probability, and its formula with solved examples in this article.
Conditional Probability
Table of Content
- What is Conditional Probability?
- Conditional Probability Definition
- Conditional Probability Formula
- How to Calculate Conditional Probability?
- Conditional Probability of Independent Events
- Conditional Probability vs Joint Probability vs Marginal Probability
- Conditional Probability and Bayes’ Theorem
- Conditional Probability Examples
- Tossing a Coin
- Drawing Cards
- Properties of Conditional Probability
- Multiplication Rule of Probability
- How to Apply the Multiplication Rule?
- Applications of Conditional Probability
- Conditional Probability Questions