Probability | Maths Grade 10 CBSE

Play with it

Theory vs reality — roll a die

Pick which faces count as a success. The theoretical probability is favourable/6. Then watch the die roll thousands of times — the experimental probability creeps right up to it.

favourable faces

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The three ideas in this chapter

When all outcomes are equally likely, the theoretical probability of an event E is:

P(E) = (number of favourable outcomes) / (total number of outcomes)

Worked example · a die

Find the probability of getting an even number when a fair die is rolled.

Total outcomes = 6 (faces 1–6).
Favourable (even) = 6 → 3 outcomes.
P(even) = 3/6 = 1/2.

Watch out: the formula only works when outcomes are equally likely. A loaded die or a biased spinner needs experimental probability, not this.

Probability is always a number between 0 and 1 (inclusive):

P(E) = 0 → the event is impossible (e.g., rolling a 7 on a die).
P(E) = 1 → the event is certain (e.g., rolling a number ≤ 6).
Everything else lies between — closer to 1 means more likely.

A useful check

The probabilities of all possible outcomes always add up to 1. If your answer is negative or above 1, you've made a mistake.

Either an event happens or it doesn't — these two cover everything, so their probabilities add to 1:

P(E) + P(not E) = 1 → P(not E) = 1 − P(E)

Worked example · the easy way

A card is drawn from 52. Find the probability it is not a king.

P(king) = 4/52 = 1/13.
P(not king) = 1 − 1/13 = 12/13.
Far quicker than counting all 48 non-kings.

Common mistake: writing P(not E) = −P(E). It's 1 − P(E) — the complement, not the negative.

Why this matters

Where you'll actually use this

Probability is the maths of uncertainty — and uncertainty is everywhere. Weather, insurance, medicine, games and AI all run on it.

Weather, insurance & risk

"70% chance of rain" is a probability. Insurers price your premium from the probability of a claim; banks set interest from the probability of default. The same favourable/total idea, scaled up with data, runs trillions of rupees of decisions.

P(E) as risk

Games, cards & fairness

Every board game, card game and lottery is governed by probability — it's how we know a game is fair (or rigged). Casinos and app developers compute exact odds for dice, cards and spinners using the very formula you just learned.

Equally likely outcomes

🩺 Medical tests

Doctors weigh the probability a test is right to decide on treatment.

🤖 AI & spam filters

Machine-learning models output probabilities — "85% likely spam" — to make decisions.

🏏 Sports predictions

"Win probability" graphics update live from the match situation using probability.

🔒 More real-world applications

Medical tests, AI, sports predictions and more — each explained with a diagram. Free to unlock.

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Check yourself

Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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