The final chapter of the book — a single, powerful formula that turns coin tosses, card draws, and dice throws into exact, calculable odds.
P(E)0 ≤ P(E) ≤ 1equally likelycomplementary eventssample space
🎲14.1 From Coin Tosses to a Formula
You've heard people say "there's a 50-50 chance" or "it's a long shot." Probability is simply the branch of maths that turns those gut feelings into exact numbers. This is the final chapter of your Class X book, and it rewards you with one of the cleanest, most reusable formulas you'll ever learn.
🪙
Think of a fair coin like a perfectly balanced seesaw.. A fair, unbiased coin has no reason to favour heads over tails — just like a seesaw with equal weights on both sides has no reason to tip either way. That's what 'equally likely outcomes' means: no outcome has a built-in advantage over the others.
In Class IX you met experimental (empirical) probability, based on what actually happened across repeated trials: P(E) = (number of trials where E happened) ÷ (total trials). That works great for coins and dice, but you can't realistically repeat launching a satellite a thousand times to estimate its failure rate! For situations where outcomes are equally likely, we can skip repeated trials entirely and calculate probability directly.
Theoretical Probability:
P(E) = (Number of outcomes favourable to E) / (Number of all possible outcomes)
📜
This definition was given by Pierre Simon Laplace in 1795, building on foundations laid by J. Cardan in the 16th century. From here on in this chapter, 'probability' means this theoretical (classical) probability — and we assume every experiment has equally likely outcomes.
SET UPA die is thrown once. Possible outcomes: 1, 2, 3, 4, 5, 6 — six equally likely outcomes.
EVENTE = 'getting a number greater than 4'. Favourable outcomes: 5, 6 → count = 2.
APPLY FORMULAP(E) = 2/6 = 1/3.
COMPLEMENTF = 'getting a number ≤ 4' has favourable outcomes 1,2,3,4 → count=4, so P(F) = 4/6 = 2/3.
⚠️Not every experiment has equally likely outcomes! A bag with 4 red balls and 1 blue ball — drawing 'red' and drawing 'blue' are NOT equally likely (red is 4× more likely), even though drawing any individual ball is equally likely. This chapter only deals with experiments where the outcomes genuinely are equally likely.
🔄14.1 Elementary Events & Why Probabilities Add Up to 1
An elementary event is an event with just one outcome — like 'getting a head' or 'drawing the yellow ball.' They're the atoms of probability: every more complex event is built from combinations of elementary events.
SET UPA bag has 1 red, 1 blue, and 1 yellow ball, all equally likely to be drawn.
LIST ELEMENTARY EVENTSY = 'yellow drawn', R = 'red drawn', B = 'blue drawn' — each has exactly 1 favourable outcome out of 3 total.
COMPUTEP(Y) = P(R) = P(B) = 1/3 each.
OBSERVEP(Y) + P(R) + P(B) = 1/3 + 1/3 + 1/3 = 1.
📐General rule:. The sum of the probabilities of all the elementary events of an experiment is always exactly 1. Think of it like slicing a whole pizza 🍕 — no matter how many slices (outcomes) you cut it into, the slices always add back up to one whole pizza.
🚫This does NOT mean every possible sum, score, or category has equal probability. If you throw two dice, there are 11 possible totals (2 through 12) — but they are NOT equally likely, so you cannot just say each has probability 1/11. You'll see exactly why later in this chapter.
🙅14.1 Complementary Events — The Power of 'Not E'
Sometimes the fastest way to find a probability is to find the probability of the opposite happening, then subtract from 1. If Sangeeta has a 0.62 chance of winning her tennis match against Reshma, you instantly know Reshma's chance without any new calculation.
For any event E:
P(E) + P(not E) = 1
P(not E) = 1 − P(E)
🌗
Think of it like a light switch.. An event and its complement are like a light switch — it's always either ON or OFF, never both, never neither. Whatever the probability of 'ON', the probability of 'OFF' fills in exactly the rest of the way to 1.
SET UPSavita's birthday can be any of 365 days. Hamida's birthday can independently be any of 365 days too.
DIFFERENT BIRTHDAYSIf Hamida's birthday differs from Savita's, there are 364 favourable days out of 365: P(different) = 364/365.
USE THE COMPLEMENTP(same birthday) = 1 − P(different) = 1 − 364/365 = 1/365 — much faster than counting 'same birthday' outcomes directly!
🪙
Two coins tossed together: outcomes (H,H), (H,T), (T,H), (T,T) — all equally likely. 'At least one head' is easiest to find via its complement: P(no head at all) = P(T,T) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4.
📏14.1 Impossible & Sure Events — The 0-to-1 Boundary
Every probability you'll ever calculate sits somewhere on a number line between two guardrails: 0 (absolutely will not happen) and 1 (absolutely will happen).
Type of event
Example
Probability
Impossible event
Getting an 8 on a single die throw
P = 0
Sure / certain event
Getting a number less than 7 on a die throw
P = 1
Ordinary event
Getting a head on a coin toss
0 < P < 1
0 ≤ P(E) ≤ 1
♠
Spades
13 cards
♥
Hearts
13 cards
♦
Diamonds
13 cards
♣
Clubs
13 cards
4 suits × 13 cards = 52 cards total
Face cards (per suit): Jack, Queen, King — 3 × 4 suits = 12 face cards
A standard deck has 52 cards in 4 suits of 13 each. Spades and clubs are black; hearts and diamonds are red. Kings, queens, and jacks are called face cards — 12 of them in total (3 per suit × 4 suits).
EVENTDraw one card from a well-shuffled deck. E = 'card is an ace'.
COUNT4 aces out of 52 cards → P(E) = 4/52 = 1/13.
COMPLEMENT SHORTCUTP(not an ace) = 1 − 1/13 = 12/13 — no need to recount 48 cards by hand.
👕
Real-world twist: a carton of 100 shirts has 88 good, 8 with minor defects, 4 with major defects. Trader Jimmy only accepts 'good' shirts: P = 88/100 = 0.88. Trader Sujatha rejects only 'major defect' shirts, so she accepts good + minor-defect shirts: P = 96/100 = 0.96. Same carton, different 'favourable outcomes' depending on who's asking!
🎯14.1 – 14.2 Two Dice Together & Chapter Summary
Roll two dice — one blue, one grey — at the same time. Each die still shows 1 through 6, but now every pair of results is a distinct outcome. (2,5) is different from (5,2), just like getting a red-then-blue in a race is different from blue-then-red.
SUM = 13No pair of two dice (max 6 each) can ever total 13 → 0 outcomes. P(sum=13) = 0/36 = 0 (impossible event).
SUM ≤ 12Every single one of the 36 outcomes has a sum of 12 or less → P(sum≤12) = 36/36 = 1 (sure event).
🧠Common trap: someone might argue "there are 11 possible sums (2 through 12), so each has probability 1/11." This is WRONG — the 11 sums are not equally likely. Sum=7 has 6 favourable pairs while sum=2 has only 1. Always go back to the 36 equally likely (blue, grey) outcomes, never to the sums directly.
Experimental probability tells you what actually happened; theoretical probability predicts what should happen based on equally-likely assumptions. As the number of trials grows very large, the two tend to converge — flip a coin 10 times and you might get 7 heads, but flip it 10,000 times and you'll land very close to 50%.
TOOL 1 Probability & Complement Calculator
Enter the number of favourable outcomes and total possible outcomes (or pick a preset scenario like coin, die, deck of cards, or a custom bag of coloured balls) to instantly compute P(E) and P(not E) side by side.
APPLY THE FORMULA
P(E) = favourable/total = 4/52 = 1/13
USE THE COMPLEMENT
P(not E) = 1 − P(E) = 1 − 1/13 = 48/52 = 12/13
P(E) = 1/13 (0.0769)
P(not E) = 12/13 (0.9231)
TOOL 2 Two-Dice Sum Simulator
Pick any target sum from 2 to 12; the tool highlights every (blue, grey) pair on the 36-outcome grid that produces it and shows the resulting probability as a fraction.
blue \ grey
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
SAMPLE SPACE
Every (blue, grey) pair is possible: 6 × 6 = 36 equally likely outcomes total.
Click a card to flip it. Use Prev/Next to move through the deck.
Card 1 / 12
Q
What is the formula for theoretical probability of an event E?
A
P(E) = (Number of outcomes favourable to E) / (Number of all possible outcomes), assuming equally likely outcomes.
§4 Quiz
Modelled on the CBSE Class 10 Section A paper — Multiple Choice and Assertion–Reason questions, each worth 1 mark. 20 questions per attempt, drawn from a pool of 40.
