Theoretical Probability Class 10: Formula, Meaning & Examples

Quick answer: In Class 10, the theoretical (classical) probability of an event E is P(E) = (number of outcomes favourable to E) ÷ (number of all possible outcomes), provided all the outcomes are equally likely. So for a fair coin, P(head) = 1/2, and for a fair die, P(getting a 3) = 1/6. You don’t have to actually toss or roll anything — you count the favourable outcomes and divide by the total.

Theoretical probability predicts how likely an event is by reasoning about equally likely outcomes, unlike experimental probability, which is measured by actually repeating an experiment many times. This section is all about the theoretical approach: count, divide, done.

Two rules follow immediately and are worth memorising up front: the probability of any event always lies between 0 and 1 (0 ≤ P(E) ≤ 1), and an event and its complement add to 1 (P(E) + P(not E) = 1). This guide explains each idea with definitions, conditions and worked examples.

What is theoretical probability? (the classical formula)

Theoretical probability (also called classical probability) of an event E is defined as:

P(E) = (Number of outcomes favourable to E) / (Number of all possible outcomes)

• Meaning in words: out of everything that could happen, what fraction of it counts as the event you want?
• Variables: “favourable outcomes” = the outcomes in which the event occurs; “all possible outcomes” = the complete list of things that can happen (the sample space).
• Condition: this formula only works when the outcomes are equally likely (see the next section).

This definition was given by Pierre Simon Laplace in 1795, and it’s the backbone of the entire chapter.

Worked Example 1 (a coin). A fair coin is tossed once. Find the probability of getting a head.

• Possible outcomes: Head, Tail → total = 2.
• Favourable outcomes (a head) = 1.
• P(head) = 1/2. Similarly P(tail) = 1/2.

Worked Example 2 (a die). A fair die is thrown once. Find the probability of getting a number greater than 4.

• Possible outcomes: 1, 2, 3, 4, 5, 6 → total = 6.
• Favourable outcomes (greater than 4): 5, 6 → 2 outcomes.
• P(number > 4) = 2/6 = 1/3.

Want to practise counting favourable outcomes without second-guessing yourself? examfront’s Topic Practice for Probability gives you graded questions right after you learn the formula.

🎬 Quick challenge: Can you find the probability of rolling a number greater than 4? 🎲

What are equally likely outcomes?

Equally likely outcomes are outcomes that each have the same chance of occurring.

• A fair (unbiased) coin: head and tail are equally likely.
• A fair die: 1, 2, 3, 4, 5, 6 are each equally likely.

Not every experiment has equally likely outcomes. If a bag has 4 red balls and 1 blue ball, drawing “a red ball” and “a blue ball” are not equally likely — red is far more probable. The classical formula P(E) = favourable/total applies only when the individual outcomes are equally likely, so always check this first.

Key condition: the classical probability formula assumes equally likely outcomes. When they aren’t equally likely, you must model the situation differently (e.g. count each ball as a separate equally likely outcome).

What is an elementary event, and why do probabilities add to 1?

An elementary event is an event that has exactly one outcome. For a die, “getting a 3” is an elementary event; “getting an even number” is not (it has three outcomes: 2, 4, 6).

A useful fact:

The sum of the probabilities of all the elementary events of an experiment is 1.

For a die: P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 6 × (1/6) = 1. This is a quick way to check your work — if your separate probabilities don’t add up to 1, something is off.

What is a complementary event? (P(not E) = 1 − P(E))

The complement of an event E, written “not E” (or E̅), is the event that E does not happen. Since either E happens or it doesn’t:

P(E) + P(not E) = 1, so P(not E) = 1 − P(E)

• Meaning in words: the chance of an event and the chance of it not happening always add to a full 1 (100%).
• Variables: P(E) is the probability of the event; P(not E) is the probability of its complement.
• Condition: always true for any event E.

Worked Example 3 (a deck of cards). One card is drawn from a well-shuffled deck of 52 cards. Find (i) the probability the card is an ace, (ii) the probability it is not an ace.

• Total outcomes = 52. There are 4 aces.
• (i) P(ace) = 4/52 = 1/13.
• (ii) P(not an ace) = 1 − P(ace) = 1 − 1/13 = 12/13. (Check: there are 48 non-aces, and 48/52 = 12/13 ✓.)

The complement rule is a huge time-saver: instead of counting 48 non-aces, subtract from 1.

The card, dice and marble problems get longer from here — practise the full graded set inside examfront’s Chapter Quizzes, and let Mistake Identification catch the counting slips before the exam does.

What is the range of probability? (sure and impossible events)

For any event E:

0 ≤ P(E) ≤ 1

Because the number of favourable outcomes can never exceed the total number of outcomes, and can never be negative. Two boundary cases give the endpoints:

• Impossible event → P = 0. Example: getting an 8 on a normal die. No face shows 8, so 0 favourable outcomes → P = 0/6 = 0.
• Sure (certain) event → P = 1. Example: getting a number less than 7 on a die. All 6 faces qualify → P = 6/6 = 1.

So a value like 1.5, −0.2, or 8/3 can never be a probability. If you ever compute one, recount.

🎬 Quick challenge: Can you find the probability of rolling a number greater than 4? 🎲

Theoretical vs. experimental probability (a quick contrast)

• Theoretical (this section): calculated by reasoning about equally likely outcomes — P(E) = favourable/total. No experiment performed.
• Experimental (empirical, from Class 9): measured by actually repeating an experiment — P(E) = (trials in which E happened) / (total trials).

As the number of trials grows large, the experimental probability tends to get close to the theoretical value — but they are found in different ways.

Key Takeaways

• Theoretical probability: P(E) = favourable outcomes ÷ total outcomes, when outcomes are equally likely.
• Outcomes must be equally likely for the classical formula to apply — always check first.
• An elementary event has one outcome; the probabilities of all elementary events sum to 1.
• Complement rule: P(not E) = 1 − P(E) — often the fastest route to an answer.
• Probability is always between 0 and 1: impossible event = 0, sure event = 1.

Quick Facts

• Probability of a sure event = 1; probability of an impossible event = 0.
• Range of any probability: 0 ≤ P(E) ≤ 1.
• P(head) on a fair coin = 1/2; P(any given number) on a fair die = 1/6.
• A standard deck has 52 cards, 4 aces, 12 face cards (K, Q, J × 4 suits), 26 red and 26 black.
• The classical definition of probability is due to Laplace (1795).

Common Mistakes

1. Using the formula when outcomes aren’t equally likely — check for equal likelihood first.
2. Miscounting the total number of outcomes (the sample space) — list them if unsure.
3. Forgetting the complement rule — P(not E) = 1 − P(E) is often quicker than direct counting.
4. Writing a probability greater than 1 or negative — impossible; recount.
5. Confusing theoretical with experimental probability — one is reasoned, the other is measured.

FAQ

What is the formula for theoretical probability in Class 10? Theoretical (classical) probability of an event E is P(E) = (number of outcomes favourable to E) ÷ (number of all possible outcomes), assuming all outcomes are equally likely. For a fair coin, P(head) = 1/2, because there is 1 favourable outcome out of 2 possible outcomes.

What is the difference between theoretical and experimental probability? Theoretical probability is calculated from the assumption of equally likely outcomes using P(E) = favourable/total, without doing the experiment. Experimental (empirical) probability is measured by actually repeating an experiment and using (trials in which the event happened) ÷ (total trials). Over many trials, the experimental value tends towards the theoretical one.

What are equally likely outcomes? Equally likely outcomes each have the same chance of occurring — like head/tail on a fair coin or 1–6 on a fair die. The classical probability formula only applies when the outcomes are equally likely.

What is a complementary event and its probability? The complement of an event E (“not E”) is the event that E does not happen. Their probabilities satisfy P(E) + P(not E) = 1, so P(not E) = 1 − P(E). If P(ace) = 1/13, then P(not an ace) = 12/13.

Can the probability of an event be more than 1 or negative? No. For any event, 0 ≤ P(E) ≤ 1. An impossible event has probability 0 and a sure event has probability 1, so values like 1.5 or −0.2 are never valid probabilities.

Related Concepts

• Experimental (empirical) probability (Class 9) — the measured approach that this theoretical approach complements.
• Sample space and outcomes — listing all possible outcomes correctly is the foundation of every probability question.
• Fractions, ratio and percentage — probabilities are written as fractions and can be read as percentages.
• Probability chapter guide — the full chapter roadmap, including two-dice and card problem types.
• Class 10 Maths formula sheet — every probability formula in one place.
• Previous year questions — practise the exact probability question types the board asks.

Ready to test yourself? Start with examfront’s Probability quiz, track which question types trip you up with Progress Tracking, and get personalised help exactly where you slip.