Probability

Why This Matters

Before a cricket match, the captains toss a coin. Why does everyone accept that as fair? Because a fair coin is just as likely to land heads as tails — neither team has an edge. That single, everyday idea is the whole heart of this chapter: when a few outcomes are equally likely, we can put an exact number on “how likely” each one is.

That number is a probability — a measure of chance that runs from 0 (never happens) to 1 (always happens). Weather forecasts, insurance premiums, the odds in a game, even whether a launched satellite is expected to fail — all of them rest on probability. Scientists, doctors and economists lean on it every day to make decisions when the future is uncertain.

In Class IX you found probabilities by doing the experiment many times and counting — tossing a coin 1000 times and seeing how often heads showed up. That’s powerful, but you can’t always repeat an experiment (you can’t relaunch a satellite a thousand times to estimate failure). This chapter takes a smarter route: when outcomes are equally likely, you can calculate the probability directly, without any experiment at all.

The Big Idea

When an experiment has a fixed list of equally likely outcomes, the theoretical (classical) probability of an event E is simply a count: P(E) = (number of outcomes favourable to E) / (total number of possible outcomes). Every probability lies between 0 and 1: an impossible event has probability 0, a sure event has probability 1. And the chance of E not happening fills in the rest: P(not E) = 1 − P(E).

Let’s Break It Down

What theoretical probability is

First, two words we’ll use a lot. An experiment is an action with more than one possible result (tossing a coin, throwing a die, drawing a card). Each possible result is an outcome. An event is any collection of outcomes we care about — like “getting an even number” on a die, which is the three outcomes 2, 4 and 6.

The whole method rests on one assumption: the outcomes are equally likely — no outcome is favoured over another. A fair coin (we say unbiased), a fair die, a well-shuffled deck — all give equally likely outcomes. When that holds, Pierre-Simon Laplace’s 1795 definition gives the probability of an event E directly:

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

That’s it — favourable divided by total. No experiment needed. For one toss of a coin, the two equally likely outcomes are Head and Tail; the event “head” has 1 favourable outcome out of 2, so P(head) = 1/2 and likewise P(tail) = 1/2.

An event with only one outcome is called an elementary event — like “getting a head”, or “the die shows 3”. A neat fact: the probabilities of all the elementary events of an experiment add up to 1. For the die, P(1) + P(2) + … + P(6) = 1/6 × 6 = 1.

The 0-to-1 range: sure and impossible events

The favourable count can never be smaller than 0 or larger than the total, so every probability is squeezed between 0 and 1:

0 ≤ P(E) ≤ 1

The two extremes have names. An impossible event can never happen, so it has 0 favourable outcomes and P = 0. Throwing a single die and getting an 8 is impossible — no face is marked 8 — so P(getting 8) = 0/6 = 0. A sure (or certain) event always happens, so every outcome is favourable and P = 1. Getting a number less than 7 on a die is certain — all six faces qualify — so P(less than 7) = 6/6 = 1.

Complementary events: P(not E) = 1 − P(E)

For any event E, there’s a partner event “not E” — everything that is not in E. We write it E̅. Together E and E̅ cover all the outcomes with no overlap, so their probabilities must add to the whole:

P(E) + P(not E) = 1, which rearranges to P(not E) = 1 − P(E).

E and E̅ are called complementary events. This little formula is a huge shortcut: often the event you want is messy to count directly, but its complement is easy. Instead of counting that, count the complement and subtract from 1.

For example, drawing a card from a deck, P(not an ace) is tedious to count outcome by outcome — but there are 4 aces in 52 cards, so P(ace) = 4/52 = 1/13, and therefore P(not an ace) = 1 − 1/13 = 12/13. Much faster.

Worked examples: coins, dice and cards

These three setups appear again and again, so it pays to know each one’s outcome list cold. A coin has 2 outcomes (Head, Tail). A die has 6 (the faces 1 to 6). A standard deck has 52 cards in 4 suits of 13 — spades and clubs are black, hearts and diamonds are red; each suit runs ace, king, queen, jack, 10, 9, …, 2, and the kings, queens and jacks are the 12 face cards.

Now the classic two-dice problem. When two dice are thrown, an outcome is an ordered pair (first die, second die), so the pair 1-then-4 is different from 4-then-1. That gives 6 × 6 = 36 equally likely outcomes — best seen as a grid:

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Summary

You should now be able to explain:

• The theoretical (classical) probability of an event E is P(E) = (number of favourable outcomes) / (total number of all possible outcomes), provided the outcomes are equally likely.
• The probability of a sure (certain) event is 1; the probability of an impossible event is 0.
• Every probability lies in the range 0 ≤ P(E) ≤ 1.
• An elementary event has just one outcome, and the probabilities of all elementary events of an experiment add up to 1.
• For any event E, P(E) + P(not E) = 1, so P(not E) = 1 − P(E); E and not-E are complementary events, and the complement is often the quick way to an answer.
• Know your outcome lists: a coin has 2 outcomes, a die has 6, two dice have 36 ordered pairs, and a deck has 52 cards (4 suits of 13, with 12 face cards).
• Experimental probability (Class IX) comes from actually repeating trials; theoretical probability predicts the chance from assumptions. As the number of trials grows, the two get closer and closer.

What’s Next

This is the last chapter of your Class 10 Maths journey — and a fitting place to look back. Notice how it pulls together threads from across the year: the fractions and ratios behind every P(E), the careful listing and counting that echoes the logic of earlier chapters, and the comfort with 0, 1 and everything between that runs through real numbers and beyond. From Real Numbers and Polynomials to Linear Equations, Quadratics, Trigonometry, Coordinate Geometry, Areas and Volumes, Statistics and now Probability, each chapter has been one more tool for describing the world precisely. Probability is special because it lets you reason confidently even when you can’t be certain — exactly the situation in most real decisions. Keep applying this maths: estimate, model, check, and predict. You now have a full toolkit — go use it.