CA Foundation Quantitative Aptitude Probability by Anurag Chauhan Sir

CA Foundation Quantitative Aptitude Probability is one of the most important chapters in the syllabus because it helps students understand how to measure uncertainty. Probability is used whenever outcomes are not fixed, such as predicting chances, risks, or future events.

In the CA Foundation exam, probability questions are mostly concept-based and calculation-oriented. Students are tested on definitions, formulas, logical thinking, and real-life applications. A clear understanding of probability rules, conditional probability, and probability distributions can help students score well with less time.

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What is Probability?

Probability is a numerical measure that tells us how likely an event is to happen. The value of probability always lies between 0 and 1.

• A probability of 0 means the event is impossible

• A probability of 1 means the event is certain

Probability was first developed to solve gambling problems but is now widely used in finance, insurance, statistics, economics, and risk analysis.

Classical Definition of Probability

According to the classical approach, probability is defined as:

Probability of an event = Number of favorable outcomes / Total number of possible outcomes

Here:

• Favorable outcomes are the results we are interested in

• Total possible outcomes form the sample space

For example:

• Probability of getting a head in a coin toss is 1/2

• Probability of getting an odd number on a die is 3/6, which simplifies to 1/2

This definition works only when all outcomes are equally likely.

Fundamental Properties of Probability

Probability follows some basic rules that always hold true:

• The value of probability always lies between 0 and 1

• An impossible event has probability 0

• A sure event has probability 1

• The probability of an event and its complement always add up to 1

If P(E) is the probability of an event, then
P(Not E) = 1 − P(E)

Objective and Subjective Probability

Probability can be divided into two types based on how it is measured.

Objective probability is based on logic, experiments, or historical data. It does not depend on personal opinion. For example, the probability of getting a head in a fair coin toss is always 1/2.

Subjective probability is based on personal judgment or belief. It can vary from person to person. For example, a doctor estimating a patient’s recovery chance uses experience, not fixed data.

Basic Terminology in Probability

A random experiment is an action that can give more than one outcome, but the exact result cannot be predicted in advance. Examples include tossing a coin or throwing a die.

The sample space is the complete set of all possible outcomes of a random experiment.
For one coin toss, the sample space is {Head, Tail}.
For one die throw, the sample space is {1, 2, 3, 4, 5, 6}.

An event is a part of the sample space. For example, getting an even number on a die is an event.

A simple event has only one outcome, while a compound event has more than one outcome.

Equally Likely, Mutually Exclusive, and Exhaustive Events

Events are equally likely if each event has the same chance of occurring, such as head or tail in a fair coin toss.

Two events are mutually exclusive if they cannot occur at the same time. If one happens, the other cannot. For mutually exclusive events, the probability of both happening together is zero.

Events are mutually exhaustive if they cover all possible outcomes of an experiment. This means at least one of the events must occur.

Basic Probability Rules with Examples

If a die is thrown once, the probability of getting a number less than 4 is calculated by counting favorable outcomes {1, 2, 3} out of total outcomes {1 to 6}. The probability comes out to be 1/2.

Understanding standard cases like coin tosses, dice, and cards is very important for exams.

Probability Using Playing Cards

A standard deck has 52 cards. There are 4 suits and 13 cards in each suit. There are 12 face cards in total (Jack, Queen, King). Ace is not a face card.

For example:

• Probability of drawing a king is 4/52, which simplifies to 1/13

• Probability of drawing a red card is 26/52, which simplifies to 1/2

Set Theory in Probability

Set theory helps in solving probability problems involving more than one event.

• “A or B” means union of events

• “A and B” means intersection of events

• “Neither A nor B” means the complement of their union

A very important formula is:
P(A or B) = P(A) + P(B) − P(A and B)

This formula avoids double counting of common outcomes.

Conditional Probability

Conditional probability means finding the probability of an event when another event has already occurred.

It is written as P(A|B), which means “probability of A given B”.

The formula is:
P(A|B) = P(A and B) / P(B), provided P(B) is not zero.

Conditional probability reduces the sample space based on the given condition.

Multiplication Theorem of Probability

The multiplication theorem is used to find the probability of two events happening together.

The general formula is:
P(A and B) = P(A) × P(B|A)

If events are independent, then the formula becomes:
P(A and B) = P(A) × P(B)

This concept is very important in problems involving successive events.

With Replacement and Without Replacement

When an object is drawn and then put back, it is called with replacement, and probabilities remain the same.

When an object is not put back, it is without replacement, and probabilities change after each draw.

If not clearly stated in the question, it is usually assumed that draws are made without replacement.

Odds in Probability

Odds express probability in the form of a ratio.

• Odds in favor = Successes : Failures

• Odds against = Failures : Successes

Odds can easily be converted into probability by dividing favorable outcomes by total outcomes.

Total Probability Theorem

The total probability theorem is used when an event can occur through different mutually exclusive paths.

The total probability is found by adding probabilities of all possible ways through which the event can occur.

This concept is widely used in real-life problems such as defective goods from different factories.

Probability Distribution

A probability distribution shows how the total probability of 1 is distributed among all possible values of a random variable.

In probability distribution:

• The sum of all probabilities is always 1

• Each probability value is non-negative

Probability distributions are the base for expectation, variance, and standard deviation.

Expectation, Variance, and Standard Deviation

Expectation is the average value of a random variable and represents its central tendency.

Variance measures how much the values of a random variable are spread out from the mean.

Variance is calculated using:
Variance = E(X²) − [E(X)]²

Standard deviation is the square root of variance and shows dispersion in the same unit as the variable.

Effect of Change of Origin and Scale

If a constant is added or multiplied to a random variable:

• Mean changes with both origin and scale

• Variance and standard deviation are affected only by scale, not origin

This concept is useful in transformation-based questions.

Discrete and Continuous Random Variables

A discrete random variable takes countable values, such as number of heads or number of defects.

A continuous random variable can take any value within a range, such as height, weight, or time.

Discrete variables use probability mass functions, while continuous variables use probability density functions.

Theoretical Approaches to Probability

The classical approach assumes equally likely outcomes.

The axiomatic approach is based on mathematical axioms and is more general.

The relative frequency approach defines probability as the limit of frequency when trials become very large.