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Fundamentals of Business Mathematics and Statistics · Probability

Mathematical Expectation: Formula, Properties and Solved Examples

Updated 10 October 2026 · Fact-checked

Mathematical expectation is the long-run average value of a random variable. You find it by multiplying each possible value by its probability and adding all the products: E(X) = Σ x·p(x). In business questions, it helps you compare options by their average payoff.

Understand Mathematical Expectation

A random variable is a quantity whose value depends on chance. For example, the number on a die, or the daily profit of a shop. Each possible value has a probability. The list of values with their probabilities is the probability distribution.

Mathematical expectation, or expected value, E(X), is the average you would get if the experiment were repeated a very large number of times. It is a weighted average. The weights are the probabilities.

The expected value need not be a value X can actually take. A fair die has E(X) = 3.5, though you can never roll 3.5. It is the long-run average, not a prediction of one trial.

In business, you use it to choose between options. If project A has a higher expected profit than project B, A is better on average. The same idea applies to expected sales, expected loss and the fair price of a game.

For a valid distribution, every probability must lie between 0 and 1, and all probabilities must add up to 1. Check this first in every question.

Key formulas to remember

Expected value (discrete)
E(X) = Σ x·p(x) = x₁p₁ + x₂p₂ + ... + xₙpₙ
Multiply each value by its probability, then add.
Valid distribution
Σ p(x) = 1 and 0 ≤ p(x) ≤ 1
Use this to find a missing probability or an unknown constant.
Expectation of a function
E(X²) = Σ x²·p(x)
Square the values only, not the probabilities.
Expectation of a constant
E(c) = c
A fixed number has no chance element.
Multiplication by a constant and shift
E(aX + b) = a·E(X) + b
Holds for any constants a and b.
Addition property
E(X + Y) = E(X) + E(Y)
Always true for any two random variables, independent or not.
Multiplication property
E(XY) = E(X)·E(Y)
Valid only when X and Y are independent.
Variance using expectation
Var(X) = E(X²) − [E(X)]²
Useful when a question asks for variance after expectation.

How to solve Mathematical Expectation questions

Use this method for any question on expected value, from a plain table to a business payoff problem.

  1. 1List all possible values of X and their probabilities in two rows or columns.
  2. 2Check that the probabilities add up to 1. If one is missing or unknown, find it from Σp = 1.
  3. 3For payoff questions, write each outcome as a rupee gain or loss. Show losses as negative numbers.
  4. 4Multiply each value by its probability to get the products x·p.
  5. 5Add all the products. This sum is E(X).
  6. 6If the question asks for E(X²), E(aX + b) or variance, apply the matching formula using E(X) already found.
  7. 7For decisions, compute the expectation of each option and choose the one with the higher expected profit or lower expected cost.

Quickest way: Table and mental sum method

When to use it: Use it when the MCQ gives a small distribution with simple probabilities such as tenths or quarters.

  1. Do not expand the formula in words. Go straight to adding x × p across the options.
  2. If probabilities share a denominator, multiply values by the numerators and divide once at the end.
  3. For E(aX + b), find E(X) first and then apply a and b. This is quicker than changing every value.
  4. For a fair die, remember E = 3.5. For a fair coin counting heads in two tosses, E = 1.
  5. Eliminate options that lie outside the smallest and largest values of X. E(X) always lies between them.

Common mistakes in Mathematical Expectation

  • Dividing the sum of x·p by the number of values again.

    Students mix this up with the ordinary arithmetic mean.

    Fix: Since probabilities already add up to 1, E(X) = Σ x·p is the final answer. Do not divide again.

  • Not checking that Σp = 1 before calculating.

    Students rush to multiply and miss that a probability is missing.

    Fix: Add the probabilities first. Find any unknown from Σp = 1.

  • Writing E(X²) as [E(X)]².

    It looks like squaring the result of the average.

    Fix: For E(X²), square each x first, multiply by p, then add. In general E(X²) is not equal to [E(X)]².

  • Using E(XY) = E(X)E(Y) without independence.

    The addition property works always, so students assume multiplication does too.

    Fix: Use the multiplication rule only when the question says X and Y are independent.

  • Ignoring the sign of losses.

    Students write the loss amount as a positive number.

    Fix: Write losses as negative values, such as −₹2,000, before multiplying by the probability.

  • Applying the constant in E(aX + b) wrongly, such as E(aX + b) = a·E(X + b).

    Confusion about what is multiplied by a.

    Fix: Remember that b is added once after multiplying: E(aX + b) = a·E(X) + b.

Worked examples

Example 1

A shopkeeper in Pune records daily sales of a product. The number of units sold X is 10, 20, 30 or 40 with probabilities 0.1, 0.3, 0.4 and 0.2 respectively. If each unit gives a profit of ₹50, find the expected daily profit.

Show the solution
  1. Check probabilities: 0.1 + 0.3 + 0.4 + 0.2 = 1.0, so the distribution is valid.
  2. Compute E(X) = 10(0.1) + 20(0.3) + 30(0.4) + 40(0.2).
  3. E(X) = 1 + 6 + 12 + 8 = 27 units.
  4. Profit = 50X, so expected profit = 50 × E(X).
  5. Expected profit = 50 × 27 = ₹1,350.

Answer: Expected daily profit = ₹1,350.

Example 2

A company in Chennai is deciding on a new product. It may earn a profit of ₹5,00,000 with probability 0.4, ₹1,00,000 with probability 0.4, or make a loss of ₹2,00,000 with probability 0.2. Find the expected profit.

Show the solution
  1. Check probabilities: 0.4 + 0.4 + 0.2 = 1.0, so the distribution is valid.
  2. Write the loss as a negative value: −₹2,00,000.
  3. Multiply: 5,00,000 × 0.4 = 2,00,000.
  4. Multiply: 1,00,000 × 0.4 = 40,000.
  5. Multiply: −2,00,000 × 0.2 = −40,000.
  6. Add: 2,00,000 + 40,000 − 40,000 = 2,00,000.

Answer: Expected profit = ₹2,00,000.

Exam tips

  • Expect short MCQs where you must compute E(X) from a table. Practise doing the sum quickly without a full table.
  • Watch for questions with an unknown probability or constant k. Find it from Σp = 1 first, then compute E(X).
  • Remember the properties as direct-use questions: given E(X), find E(aX + b) in one line.
  • In decision questions, compare expected values of the options and pick the best. Do not be distracted by the largest single payoff.
  • Since there is no negative marking, always attempt every question. Use the range check to eliminate options that cannot be an expected value.

Practice questions from Probability

Mathematical Expectation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Mathematical Expectation: frequently asked questions

What is the formula for mathematical expectation?

For a discrete random variable, E(X) = Σ x·p(x). You multiply each value by its probability and add all the products. The probabilities must add up to 1.

Can expected value be a number that X can never take?

Yes. The expected value is a long-run average, not a single outcome. For a fair die, E(X) = 3.5, though no roll gives 3.5.

Is E(X + Y) = E(X) + E(Y) always true?

Yes, the addition property holds for any two random variables, whether or not they are independent. The multiplication property E(XY) = E(X)E(Y) needs independence.

How is expectation used in business decisions?

You compute the expected profit or cost of each option using the probabilities of different outcomes. You then choose the option with the better expected value. It gives a fair comparison under uncertainty.