Probability: Formulas, Concepts and Aptitude Questions
Probability measures the chance of an event occurring. This chapter covers sample spaces, events, complementary events, mutually exclusive and independent events, conditional probability, and standard addition and multiplication rules. It also includes probability formulas, shortcuts, and numerical examples used in quantitative aptitude questions.
What Is Probability?
For equally likely outcomes, probability is calculated as P(E) = Number of favourable outcomes ÷ Total number of outcomes. For example, when a fair die is rolled, the probability of getting an even number is 3/6 = 1/2 because the favourable outcomes are 2, 4, and 6.
Probability Video Lessons
Watch short topic-wise lessons for quick revision.
Complementary and Mutually Exclusive Events
Understand complementary and mutually exclusive events in probability, including their definitions, differences, and how they relate to possible outcomes in an experiment.
Probability Formula & Tricks
Important Formulas
n(E) is the number of favourable outcomes and n(S) is the total number of outcomes in the sample space.
E' represents the event that E does not occur.
Use this rule for the probability of A or B when the events may overlap.
If two events cannot occur together, their intersection has probability zero.
P(B | A) is the probability of B when A has already occurred.
For independent events, the occurrence of one event does not change the probability of the other.
This gives the probability of B under the condition that A has occurred.
Quick Tricks
For repeated independent trials, calculate the probability of none occurring first, then subtract it from 1.
The formula favourable outcomes divided by total outcomes applies directly only when all outcomes have equal chances.
When finding P(A or B), subtract the intersection once because it is included in both P(A) and P(B).
Probability Concepts
Random Experiment, Sample Space and Event
For a single coin toss, the sample space is S = {H, T}. For one die roll, S = {1, 2, 3, 4, 5, 6}. An event such as getting an odd number is E = {1, 3, 5}. The number of outcomes in a set is denoted by n(E).
Classical Probability and Range
An impossible event has probability 0, while a certain event has probability 1. If E and E' are complementary events, then P(E) + P(E') = 1. Fractions, decimals, and percentages can all represent probability.
Addition Rule and Mutually Exclusive Events
Mutually exclusive events have no common outcome. Events that can occur together are not mutually exclusive, so their common outcomes must be subtracted once to prevent double counting.
Multiplication Rule and Conditional Probability
When A has already occurred, only outcomes satisfying A remain relevant. If A and B are independent, P(B | A) = P(B), so the multiplication rule becomes P(A ∩ B) = P(A)P(B).
Independent and Dependent Events
Successive coin tosses are independent because earlier tosses do not change the next toss. Drawing cards without replacement creates dependent events because the first draw changes the remaining cards. For independent events, P(A ∩ B) = P(A)P(B).
Probability Data Table
Use the structured data below while following the explanation and examples.
Common Probability Events
Basic event types and their probability properties.
| Event type | Meaning | Probability property |
|---|---|---|
| Impossible event | Cannot occur | P(E) = 0 |
| Certain event | Must occur | P(E) = 1 |
| Complementary event | Event does not occur | P(E') = 1 - P(E) |
| Mutually exclusive events | Cannot occur together | P(A ∩ B) = 0 |
| Independent events | One does not affect the other | P(A ∩ B) = P(A)P(B) |
Note: Mutually exclusive and independent are different conditions. Two events with positive probabilities cannot be both mutually exclusive and independent.
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Probability Revision Points
Key definitions, formulas, and rules for quick revision.
- Probability always lies between 0 and 1.
- For equally likely outcomes, P(E) = favourable outcomes ÷ total outcomes.
- P(E') = 1 - P(E).
- P(A or B) = P(A) + P(B) - P(A and B).
- For mutually exclusive events, P(A and B) = 0.
- P(A and B) = P(A)P(B | A).
- For independent events, P(A and B) = P(A)P(B).
- Use 1 - P(none) to find the probability of at least one occurrence.
Probability FAQs
What is the probability of an impossible event?
The probability of an impossible event is 0.
What is the probability of a certain event?
The probability of a certain event is 1.
What is the probability of getting at least one head in three fair coin tosses?
P(at least one head) = 1 - P(no heads) = 1 - (1/2)^3 = 7/8.
How is probability calculated when events overlap?
Use P(A ∪ B) = P(A) + P(B) - P(A ∩ B). The intersection is subtracted once to remove double counting.
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events do not affect each other, so P(A ∩ B) = P(A)P(B).
What is the probability of drawing two aces consecutively without replacement?
It is 4/52 × 3/51 = 1/221. The events are dependent because the first card is not replaced.