Preparing for the AAI ATC 2026 examination requires a strong understanding of mathematical concepts, formulas, and problem-solving techniques. Candidates should focus on learning the fundamentals, practising numerical questions, and developing the ability to solve multiple-choice questions accurately under time pressure.

This guide covers three important mathematics topics: Linear Programming, Conditional Probability and Bayes’ Theorem, and Binomial Probability Distribution. Each section includes key concepts, formulas, solved examples, and practice questions to help students revise efficiently.

Whether you are beginning your preparation or revising before a mock test, these notes from Aviate Learnings can help you organise your study sessions and strengthen your mathematical fundamentals.

1. Linear Programming: Concepts, Formulas and Solved Examples

Linear Programming is a mathematical technique used to find the maximum or minimum value of a linear objective function while satisfying a set of constraints.

It is commonly associated with optimisation problems involving limited resources, production planning, transportation, and cost management.

For mathematics preparation, the most important skills are converting a word problem into mathematical expressions, drawing constraints graphically, identifying the feasible region, and evaluating the objective function.

Important terms in Linear Programming

Objective function: The linear expression whose maximum or minimum value must be found.

For example:

Z = 3x + 2y

Here, Z is the objective function, while x and y are decision variables.

Constraints: Conditions that restrict the possible values of the decision variables.

Examples include:

x + y ≤ 10

2x + y ≤ 14

x ≥ 0, y ≥ 0

Feasible region: The common region containing all points that satisfy every constraint simultaneously.

Optimal solution: A feasible solution that gives the required maximum or minimum value of the objective function.

Steps to solve a Linear Programming problem

1. Identify the decision variables.
2. Form the objective function.
3. Convert the conditions into linear inequalities.
4. Include non-negativity constraints when required.
5. Draw the constraint lines on a graph.
6. Identify the common feasible region.
7. Find the coordinates of its corner points.
8. Calculate the objective function at each corner point.
9. Compare the values and select the maximum or minimum as required.

For a standard graphical linear programming problem with a non-empty, bounded feasible region, an optimal value occurs at a corner point. If the feasible region is unbounded, an optimum may or may not exist, so the objective function and constraints must be examined carefully.

Solved Example 1: Maximisation

Maximise:

Z = 3x + 2y

Subject to:

x + y ≤ 8

x + 2y ≤ 10

x ≥ 0, y ≥ 0

The corner points of the feasible region are:

(0, 0), (8, 0), (6, 2), and (0, 5).

Evaluate Z at each point.

Corner point	Z = 3x + 2y	(0, 0)	0	(8, 0)	24	(6, 2)	22	(0, 5)	10

The maximum value is 24 at (8, 0).

Answer: Maximum Z = 24.

Solved Example 2: Evaluating the objective function

Suppose the objective function is:

Z = 4x + 3y

Find Z at the point (2, 2).

Substitute x = 2 and y = 2:

Z = 4(2) + 3(2)

Z = 8 + 6

Answer: Z = 14.

Remember that evaluating the objective function at a point does not automatically make that point an optimal solution. The point must first satisfy all the constraints, and its value must be compared with the other relevant corner points.

Common mistakes in Linear Programming

• Drawing a constraint line with an incorrect intercept.
• Shading the wrong half-plane.
• Forgetting non-negativity conditions.
• Evaluating the objective function at only one corner point.
• Confusing the feasible region with the area outside it.
• Assuming every unbounded problem has a maximum or minimum.

Practise graphical questions regularly because understanding the feasible region is essential for solving optimisation problems correctly.

2. Conditional Probability: Definition and Important Formulas

Conditional Probability measures the probability of an event occurring when another event is already known to have occurred.

It is useful when information about one event changes the probability of another event. Examples include drawing objects without replacement, analysing test results, and evaluating events under specific conditions.

Let A and B be two events, with P(B) greater than zero.

The formula for conditional probability is:

P(A | B) = P(A ∩ B) / P(B)

Here:

• P(A | B) is the probability of A occurring given that B has occurred.
• P(A ∩ B) is the probability that both A and B occur.
• P(B) is the probability of event B.

The order of the events matters. In general, P(A | B) is not equal to P(B | A).

Multiplication rule of probability

Rearranging the conditional probability formula gives:

P(A ∩ B) = P(A | B) × P(B)

Similarly:

P(A ∩ B) = P(B | A) × P(A)

These expressions are particularly useful in questions involving two events.

Independent events

Two events are independent if the occurrence of one does not change the probability of the other.

For independent events A and B:

P(A | B) = P(A), provided P(B) is positive.

The multiplication rule becomes:

P(A ∩ B) = P(A) × P(B)

For example, tossing a fair coin twice produces independent outcomes because the result of the first toss does not affect the probability of the second toss.

Solved Example 1: Conditional probability

A bag contains five red balls and three blue balls. Two balls are drawn successively without replacement.

Given that the first ball drawn is red, find the probability that the second ball is red.

After one red ball has been removed, four red balls remain among seven balls.

Therefore:

P(second red | first red) = 4/7

Answer: 4/7.

Notice that the probability changes because the first ball is not replaced.

Solved Example 2: Using the conditional probability formula

Suppose:

P(A ∩ B) = 0.2

P(B) = 0.5

Find P(A | B).

Using the formula:

P(A | B) = P(A ∩ B) / P(B)

P(A | B) = 0.2 / 0.5

Answer: P(A | B) = 0.4.

The key is to divide the joint probability by the probability of the event after the vertical bar.

3. Bayes’ Theorem: Formula and Applications

Bayes’ Theorem helps calculate the probability of a possible cause when an outcome or piece of evidence is already known.

It is especially useful when an outcome could have been produced by several different sources. Examples include medical testing, manufacturing defects, quality control, and reliability analysis.

Suppose B₁, B₂, …, Bₙ are mutually exclusive and exhaustive events, and A is an event with P(A) greater than zero.

Bayes’ Theorem is:

P(Bₖ | A) = [P(Bₖ) × P(A | Bₖ)] / P(A)

Using the theorem of total probability, the denominator can be expanded:

P(Bₖ | A) = [P(Bₖ) × P(A | Bₖ)] / Σ[P(Bᵢ) × P(A | Bᵢ)]

The summation includes all possible causes Bᵢ.

Solved Example: Manufacturing defects

A factory has two machines.

• Machine M₁ produces 60% of the items, with a defect rate of 2%.
• Machine M₂ produces 40% of the items, with a defect rate of 5%.

An item selected at random is found to be defective. Find the probability that it was produced by M₁.

Let D represent the event that an item is defective.

Given:

P(M₁) = 0.6

P(M₂) = 0.4

P(D | M₁) = 0.02

P(D | M₂) = 0.05

Using Bayes’ Theorem:

P(M₁ | D) = [P(M₁) × P(D | M₁)] / P(D)

First, calculate the total probability of a defective item:

P(D) = 0.6 × 0.02 + 0.4 × 0.05

P(D) = 0.012 + 0.020 = 0.032

Therefore:

P(M₁ | D) = 0.012 / 0.032

Answer: P(M₁ | D) = 0.375, or 37.5%.

This example demonstrates an important principle: the probability that an item came from a particular machine depends on both the machine’s production share and its defect rate.

Conditional Probability versus Bayes’ Theorem

Conditional probability calculates the probability of one event given another event. Bayes’ Theorem uses conditional probabilities and prior probabilities to calculate the probability of a possible cause given an observed result.

When solving a question, identify what is already known and what probability is being requested. This helps you decide whether the basic conditional probability formula or Bayes’ Theorem is more appropriate.

4. Binomial Probability Distribution: Complete Formula Guide

The Binomial Probability Distribution describes the number of successes in a fixed number of independent trials when each trial has two possible outcomes and the probability of success remains constant.

Examples include counting the number of heads in repeated coin tosses, the number of successful attempts in a fixed set of trials, and the number of defective products in a sample when the model’s assumptions hold.

A random variable X follows a binomial distribution with parameters n and p when:

• There are a fixed number of trials, n.
• Each trial has two possible outcomes, success or failure.
• Trials are independent.
• The probability of success, p, is constant.
• The probability of failure is q = 1 − p.

Binomial probability formula

The probability of obtaining exactly r successes in n trials is:

P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ

Where:

• n is the number of trials.
• r is the required number of successes.
• p is the probability of success.
• q is the probability of failure.
• ⁿCᵣ is the number of ways to choose r successes from n trials.

The combination formula is:

ⁿCᵣ = n! / [r!(n − r)!]

Mean, variance and standard deviation

The mean or expected value is:

E(X) = np

The variance is:

Var(X) = npq

The standard deviation is:

σ = √npq

These three formulas are particularly useful for short numerical questions.

Solved Example 1: Exactly three heads

A fair coin is tossed five times. Find the probability of getting exactly three heads.

Given:

n = 5

p = 0.5

q = 0.5

r = 3

Using the binomial formula:

P(X = 3) = ⁵C₃(0.5)³(0.5)²

Since ⁵C₃ = 10:

P(X = 3) = 10 × 0.125 × 0.25

Answer: P(X = 3) = 0.3125.

Solved Example 2: Finding the mean and variance

A binomial random variable has n = 20 and p = 0.4.

Calculate the mean and variance.

First, calculate the probability of failure:

q = 1 − 0.4 = 0.6

Mean:

E(X) = np = 20 × 0.4

Mean = 8.

Variance:

Var(X) = npq = 20 × 0.4 × 0.6

Variance = 4.8.

The standard deviation is √4.8, approximately 2.19.

Exact probability, at least, and at most

Pay attention to the wording of binomial questions.

Exactly r successes: Use P(X = r).

At most r successes: Add probabilities from zero through r.

P(X ≤ r) = Σ P(X = k), for k = 0 to r.

At least r successes: Add probabilities from r through n.

P(X ≥ r) = Σ P(X = k), for k = r to n.

The complement rule can sometimes save time:

P(X ≥ 1) = 1 − P(X = 0)

This is useful when calculating the probability of at least one success.

Shape of the binomial distribution

The shape of a binomial distribution depends on n and p.

• When p = 0.5, the distribution is symmetric.
• When p is below 0.5, it is generally skewed towards lower success counts.
• When p is above 0.5, it is generally skewed towards higher success counts.

For sufficiently large n, the binomial distribution may be approximated by a normal distribution when the relevant conditions are satisfied. A common introductory guideline is that both np and nq should be at least about five, although the adequacy of the approximation depends on the situation.

5. Practice MCQs for AAI ATC Maths 2026

Try answering these questions before checking the answer key.

Question 1. In a standard graphical Linear Programming problem with a bounded feasible region, the optimal value is found at:

A. Any point outside the feasible region
B. A corner point of the feasible region
C. Only the origin
D. The midpoint of every constraint line

Question 2. The common region satisfying all the constraints is called:

A. Objective region
B. Feasible region
C. Critical point
D. Intercept region

Question 3. If Z = 3x + 2y, find Z at (2, 4).

A. 10
B. 12
C. 14
D. 16

Question 4. If P(A ∩ B) = 0.2 and P(B) = 0.5, find P(A | B).

A. 0.2
B. 0.3
C. 0.4
D. 0.5

Question 5. If A and B are independent, then:

A. P(A | B) = P(A), when P(B) > 0
B. P(A ∩ B) = 0
C. P(A) = P(B) always
D. P(A | B) = 1

Question 6. Bayes’ Theorem is primarily used to find:

A. The mean of a distribution
B. The probability of a possible cause given observed evidence
C. The slope of a line
D. The maximum value of a linear function

Question 7. The mean of a binomial distribution is:

A. np
B. npq
C. √npq
D. n + p

Question 8. If n = 10 and p = 0.3, what is the mean of the binomial distribution?

A. 2
B. 3
C. 7
D. 10

Question 9. If p = 0.7, the probability of failure q is:

A. 0.3
B. 0.7
C. 1.7
D. 0

Question 10. The variance of a binomial distribution with n = 10 and p = 0.5 is:

A. 2.5
B. 5
C. 10
D. 0.5

Answer key with explanations

1. B — For a bounded feasible region with a feasible linear objective, an optimum occurs at a corner point.
2. B — The feasible region satisfies every constraint.
3. C — Z = 3(2) + 2(4) = 14.
4. C — P(A | B) = 0.2/0.5 = 0.4.
5. A — Independence means knowing B occurred does not change the probability of A.
6. B — Bayes’ Theorem updates the probability of a cause using observed evidence.
7. A — The mean is np.
8. B — Mean = 10 × 0.3 = 3.
9. A — q = 1 − p = 0.3.
10. A — Variance = 10 × 0.5 × 0.5 = 2.5.

6. How to Prepare Maths for AAI ATC 2026

A consistent study plan can help you improve your understanding and reduce calculation mistakes.

Revise one topic at a time

Begin with Linear Programming and understand how to form constraints and identify the feasible region. Move to Conditional Probability and Bayes’ Theorem, then practise Binomial Probability Distribution.

Avoid trying to memorise every formula in a single session.

Make a formula notebook

Write each formula with a short explanation of its variables and the conditions under which it applies. For example, distinguish P(A | B) from P(B | A), and distinguish the mean np from the variance npq.

Practise without looking at the solution

Attempt numerical questions independently before reviewing worked examples. If you make an error, identify whether it occurred because of the formula, substitution, arithmetic, or interpretation of the question.

Use timed practice

Once you understand the concepts, solve mixed MCQs under time limits. This helps you practise selecting the appropriate method without spending too long on a single problem.

Verify the official syllabus

Use the official AAI recruitment syllabus page to confirm the syllabus and examination details applicable to your recruitment cycle. These notes are for mathematical revision and should be used alongside the prescribed syllabus and standard textbooks.

7. Frequently Asked Questions

Q1. Which topics are covered in this AAI ATC Maths guide?

This guide covers Linear Programming, Conditional Probability, Bayes’ Theorem, and Binomial Probability Distribution, including formulas, solved examples, and practice MCQs.

Q2. What is the most important formula in Linear Programming?

There is no single formula that solves every Linear Programming problem. You need to form the objective function and constraints, identify the feasible region, and evaluate the objective function at relevant corner points.

Q3. What is the difference between conditional probability and independent events?

Conditional probability measures the chance of an event given that another event has occurred. Independent events are events for which the occurrence of one does not change the probability of the other.

Q4. When should Bayes’ Theorem be used?

Use Bayes’ Theorem when you know the probability of evidence under different possible causes and need to find the probability of a cause after observing that evidence.

Q5. What are the mean and variance of a binomial distribution?

For a binomial distribution with n trials, success probability p, and failure probability q = 1 − p, the mean is np and the variance is npq.

Q6. How can I improve my Maths score for AAI ATC?

Understand concepts first, maintain a formula notebook, solve practice MCQs, review mistakes, and practise under timed conditions. Check the official syllabus to prioritise the topics relevant to your examination.

Conclusion

Linear Programming, Conditional Probability, Bayes’ Theorem, and Binomial Probability Distribution offer several opportunities to practise mathematical reasoning, formulas, and numerical problem-solving.

Start by understanding the concepts, move on to worked examples, and then attempt the MCQs without checking the answers. Regular revision will help you recognise question patterns, choose the correct formula, and solve problems more confidently.

For more revision sheets, chapter-wise notes, and exam preparation resources, visit the Aviate Learnings AAI ATC Resource Hub.

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