What is the correct formula for the probability of an event E, given a sample space S?

n(E) / n(S). Let n(S) be the total number of possible outcomes in the sample space. Let n(E) be the number of outcomes favourable to event E. The probability of E is then defined as P(E) = n(E) / n(S). This formula assumes every outcome in S is equally likely.

For any event E, what is the valid range of P(E)?

0 to 1, inclusive. The favourable count n(E) can never be fewer than 0 or more than n(S). So the ratio n(E)/n(S) always lies between 0 and 1. A probability of 0 means the event is impossible, and 1 means it is certain. So for any event E, 0

An unbiased die is thrown once. Find the probability of getting a number greater than 4.

1/3. A standard die has six faces, numbered 1 to 6. So the sample space is S = {1, 2, 3, 4, 5, 6}, and n(S) = 6. We want faces that are a number greater than 4. The faces that are a number greater than 4 are {5, 6}. So n(E) = 2. Using P(E) = n(E)/n(S), P(E) = 2/6 = 1/3.

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is a club.

1/4. A well-shuffled pack has 52 cards, so n(S) = 52. The pack has 4 suits of 13 cards each, with 26 red cards and 26 black cards. Among these, 13 cards are a club. So n(E) = 13. Using P(E) = n(E)/n(S), P(E) = 13/52 = 1/4.

The probability that a certain event E will occur is 3/8. Find the probability that E will not occur.

5/8. We are told P(E) = 3/8. The events E and 'not E' are complementary, so together they cover the entire sample space. This means P(E) + P(not E) = 1. So P(not E) = 1 - P(E) = 1 - 3/8 = 5/8. Check: P(E) + P(not E) = 3/8 + 5/8 = 1.

Two unbiased dice are thrown together. Find the probability of getting a total divisible by 3.

1/3. Each die has 6 faces, and the two dice are thrown together. So the sample space has n(S) = 6 x 6 = 36 equally likely outcomes. We want the outcomes where the dice show a total divisible by 3. Checking each of the 36 outcomes against this condition gives n(E) = 12 favourable outcomes. Using P(E) = n(E)/n(S), P(E) = 12/36 = 1/3.

A bag contains 5 red, 9 blue balls. Two balls are drawn at random. Find the probability that both are blue.

36/91. The bag has 5 red balls, 9 blue balls, which is 14 balls in total. We draw 2 balls at random, so n(S) = 14C2 = 91, the number of ways to choose any 2 balls. We want both balls to be blue, and there are 9 blue balls in the bag. The number of ways to pick 2 blue balls is 9C2 = 36. So n(E) = 36. Using P(E) = n(E)/n(S), P(E) = 36/91 = 36/91.

A bag contains 9 white and 8 black balls. Two balls are drawn at random. Find the probability that they are of the same colour.

8/17. The bag has 9 white balls and 8 black balls, so 17 balls in total. We draw 2 balls at random, so n(S) = 17C2 = 136, the number of ways to choose any 2 balls. We want both balls to be the same colour, either both white or both black. The number of ways to pick 2 white balls is 9C2 = 36. The number of ways to pick 2 black balls is 8C2 = 28. Adding these gives n(E) = 36 + 28 = 64. Using P(E) = n(E)/n(S), P(E) = 64/136 = 8/17.

Probability: Formulas, Shortcuts and Solved Examples

Probability questions measure how likely an event is, as a number from 0 to 1.

This lesson gives each rule, the reason it works and seven worked examples. It also lists the traps that cost marks.

The idea in brief

Basic rule: favourable ÷ total.
Not E: 1 − P(E).
A or B: add, then subtract the overlap.
A then B: multiply along the branch.
Two dice: 36 outcomes.

What Probability Questions Ask

What is probability?

Probability tells you how likely an event is. It is a number from 0 to 1.

0 means impossible. 1 means certain. 1/2 means even chances.

Which words do I need?

Experiment: an act with uncertain results, such as a throw of a die.
Sample space: the set of all possible outcomes.
Event: the outcomes you want.
Favourable outcomes: the ones inside your event.

Two dice give 36 equal outcomes. Counting the green cells gives the chance of a total divisible by 3.

What do I need to know about a pack of cards?

A pack has 52 cards in 4 suits. Each suit has 13 cards.

Spades and clubs are black. Hearts and diamonds are red.

There are 12 face cards: the jack, queen and king of each suit.

Check 1

A probability can be greater than 1.

Core Rules and Why They Work

What is the basic formula?

P(E) = n(E) / n(S).

Count the favourable outcomes. Divide by all equally likely outcomes.

Why it works: each outcome has the same chance. So the share of favourable ones is the chance.

How do I find the chance of “not E”?

P(not E) = 1 − P(E).

Why it works: E either happens or it does not. The two chances make 1.

How do I find “A or B”?

P(A or B) = P(A) + P(B) − P(A and B).

Why it works: outcomes in both A and B get counted twice when you add. So subtract them once.

Draw without replacement. The second chance changes because one ball is already gone.

How do I find “A then B”?

P(A then B) = P(A) × P(B after A).

Why it works: B only gets its chance on the branches where A happened first.

How do I use combinations for draws?

For a draw of two balls at once, use n(E) = C(favourable, 2) and n(S) = C(total, 2).

Why it works: order does not matter when balls are drawn together.

Check 2

The chance of an impossible event is 0.

Seven Solved Examples

Easy, step by step: a die is thrown once. Find the chance of a number greater than 4.

Step 1: the sample space is 1, 2, 3, 4, 5, 6. That is 6 outcomes.

Step 2: the favourable numbers are 5 and 6. That is 2 outcomes.

Step 3: P = 2/6 = 1/3.

The answer is 1/3.

Easy, step by step: one card is drawn from a pack of 52. Find the chance it is a club.

Step 1: there are 52 equally likely cards.

Step 2: there are 13 clubs.

Step 3: P = 13/52 = 1/4.

The answer is 1/4.

Easy, step by step: the chance that E happens is 3/8. Find the chance that E does not happen.

Step 1: use P(not E) = 1 − P(E).

Step 2: 1 − 3/8 = 5/8.

The answer is 5/8.

Medium, step by step: two dice are thrown. Find the chance that the total is divisible by 3.

Step 1: there are 6 × 6 = 36 outcomes.

Step 2: total 3 has 2 ways, total 6 has 5, total 9 has 4 and total 12 has 1.

Step 3: 2 + 5 + 4 + 1 = 12 favourable outcomes. P = 12/36 = 1/3.

The answer is 1/3.

Medium, step by step: a bag has 5 red and 9 blue balls. Two are drawn. Find the chance both are blue.

Step 1: there are 14 balls in all. The total pairs are C(14,2) = 91.

Step 2: the blue pairs are C(9,2) = 36.

Step 3: P = 36/91.

The answer is 36/91.

Exam level, step by step: a bag has 9 white and 8 black balls. Two are drawn. Find the chance they are the same colour.

Step 1: the total pairs are C(17,2) = 136.

Step 2: white pairs are C(9,2) = 36. Black pairs are C(8,2) = 28.

Step 3: favourable = 36 + 28 = 64. P = 64/136 = 8/17.

The answer is 8/17.

Exam level, step by step: a die is thrown. Find the chance of an even number or a number greater than 4.

Step 1: even numbers are 2, 4, 6, so P = 3/6.

Step 2: numbers over 4 are 5, 6, so P = 2/6. Both are true only for 6, so P = 1/6.

Step 3: P = 3/6 + 2/6 − 1/6 = 4/6 = 2/3.

The answer is 2/3.

Check 3

In example 5, the draw uses combinations because order does not matter.

Common Traps and Shortcuts

Which traps cost marks?

Counting outcomes that are not equally likely.
Forgetting to subtract the overlap in “A or B”.
Not reducing the numbers when drawing without replacement.
Counting 2 dice as 12 outcomes instead of 36.
Leaving the answer as a decimal when the options are fractions.

Which shortcuts save time?

Use “1 − P(not E)” when the event has many cases.
For same-colour draws, add the colour pairs.
Write the two-dice table for any sum question.
Cancel the fraction at the end, not earlier.

What patterns come up in exams?

Five patterns repeat. One die or coin, two dice, cards, balls from a bag and the “or” and “not” rules.

Each takes about a minute once the sample space is clear.

Check 4

For two dice, the sample space has 36 outcomes.

0/0 understanding checks correct

Was this helpful?

Take a Quantitative Aptitude QuizRandom practice question from this subject →

Practice 76 MCQs on this topic

Set 1 Set 2 Set 3