Formulas · Mathematics

Probability

4 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 9

Experimental probability (relative frequency)

P = \frac{f}{n}

What each letter means
$P$	the experimental probability of the event, a number from 0 to 1
$f$	the number of times the event happened
$n$	the total number of trials

Do the experiment many times and count: the fraction of trials in which the event happened is its experimental probability, also called its relative frequency.

Why it works

Each trial either shows the event or does not, so the count f is between 0 and n, and f ÷ n is between 0 and 1. Over many trials this fraction settles down near the true likelihood of the event.

When to use it

When outcomes are not equally likely, or the likelihood is unknown: a drawing pin landing point up, a team winning, a machine making a faulty part. You need real data.

How to use it

Count the trials (n).
Count how many times the event happened (f).
Divide f by n; write it as a fraction, a decimal or a percentage.

To remember: How often, out of how many tries.

Other forms

Rearranged $f = P n$ How many times to expect the event in n trials.
Same formula, another formAs a percentage, the experimental probability is 100 × f ÷ n.

Worked examples

Story

A cricket team won 18 of its last 30 matches. What is the experimental probability that it wins a match?

$P = \frac{18}{30} = \frac{3}{5} = 0.6$

Answer: 0.6

Picture

A circle is drawn so that it just fits inside a square board. Darts are thrown at the board at random, and 157 of 200 darts land inside the circle. Estimate the probability of landing inside the circle.

$P = \frac{157}{200} = 0.785$
This is close to the circle's share of the square's area, π ÷ 4 ≈ 0.785.

Answer: 0.785

Direct

A die is rolled 50 times and lands on 4 exactly 8 times. What is the experimental probability of rolling a 4?

$P = \frac{8}{50} = 0.16$
That is 16%.

Answer: 0.16

Reverse

A coin shows heads with an experimental probability of 0.45, and heads came up 36 times. How many times was the coin tossed?

Try it first, then show the working

$0.45 = \frac{36}{n}$
$n = \frac{36}{0.45} = 80$

Answer: 80

Exam

A die is rolled 600 times and shows a six 94 times. Find the experimental probability of a six and compare it with the theoretical probability 1/6.

Try it first, then show the working

$\frac{94}{600} = \frac{47}{300}$
That is about 0.157; the theoretical value 1/6 is about 0.167.
They are close, and they would be expected to get closer with more rolls.

Answer: 47/300, about 0.157, close to 1/6

Common mistake: Dividing by the number of times the event did not happen, instead of by all the trials; or trusting a result from only a few trials.

Class 9

Theoretical probability

P = \frac{m}{n}

What each letter means
$P$	the probability of the event, a number from 0 to 1
$m$	the number of favourable outcomes (the outcomes where the event happens)
$n$	the number of possible outcomes, all equally likely

When every outcome is equally likely, the probability of an event is the number of outcomes that give it, out of all the outcomes.

Why it works

If n outcomes are equally likely, each one has probability 1 ÷ n, since together they must make 1 (something certainly happens). An event made of m of them has m of those shares: m ÷ n.

When to use it

Fair coins, fair dice, well-shuffled cards, spinners with equal parts, picking at random from a bag or a list: anywhere the outcomes are equally likely and can be counted.

How to use it

List or count all possible outcomes (the sample space); check they are equally likely.
Count the outcomes where the event happens.
Divide favourable by possible.

To remember: Favourable over possible.

Other forms

Same formula, another formThe probability that the event does not happen is 1 − P, because the outcomes where it happens and where it does not together make up all n.
Special case $P = 1$ When every outcome is favourable (m = n): the event is certain.
Special case $P = 0$ When no outcome is favourable (m = 0): the event is impossible.

Worked examples

Story

A letter is picked at random from the word PROBABILITY. What is the probability that it is a B?

PROBABILITY has 11 letters, and 2 of them are B.
$P = \frac{2}{11}$
That is about 0.182, or 18.2%.

Answer: 2/11

Picture

A spinner is a circle divided into 8 equal sectors, and 3 of them are red. What is the probability that it stops on red?

The 8 equal sectors are equally likely, and 3 are red.
$P = \frac{3}{8} = 0.375$

Answer: 3/8

Direct

A fair die is rolled. What is the probability of getting a 4?

One outcome (4) is favourable out of six (1 to 6).
$P = \frac{1}{6}$
That is about 0.167, or 16.7%.

Answer: 1/6

Reverse

A bag holds 20 marbles. The probability of picking a blue one at random is 3/5. How many blue marbles are there?

Try it first, then show the working

$m = \frac{3}{5} \times 20 = 12$

Answer: 12

Exam

A fair coin is tossed twice. Find the probability of exactly one head, and of at least one head.

Try it first, then show the working

The tree diagram gives 4 equally likely outcomes: HH, HT, TH and TT.
Exactly one head: HT and TH, so the probability is 2 ÷ 4 = 1/2.
At least one head: every outcome except TT, so it is 1 − 1/4 = 3/4.

Answer: 1/2 and 3/4

Common mistake: Counting outcomes that are not equally likely as if they were (the total of two dice has 11 values, but 7 is far likelier than 2), or dividing favourable by unfavourable.

Class 10

The probability that an event does not happen: P(not E) = 1 − P(E)

Q = 1 - P

What each letter means
$Q$	the probability that the event does not happen, P(not E)
$P$	the probability that the event happens, P(E)

An event and its opposite ("not E") between them cover every outcome, so their probabilities add to 1. Knowing one gives the other.

Why it works

Every outcome either makes E happen or does not, never both. The probabilities of all the outcomes add to 1, so P(E) + P(not E) = 1, and P(not E) = 1 − P(E).

When to use it

When "not E" is easier to count than E (at least one, not a six, different birthdays), or when a game has exactly two results and you know one.

How to use it

Find the probability of whichever is easier: the event or its opposite.
Subtract it from 1 to get the other.

To remember: Happens plus does not happen makes one.

Other forms

Same formula, another formThe book's form: P(E) + P(not E) = 1.
Special case $Q = 0$ When E is certain (P = 1): its opposite is impossible.

Worked examples

Story

Sangeeta and Reshma play a tennis match, which cannot end in a draw. The probability that Sangeeta wins is 0.62. What is the probability that Reshma wins?

Reshma wins exactly when Sangeeta does not.
$Q = 1 - 0.62 = 0.38$

Answer: 0.38

Picture

One card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is not an ace?

There are 4 aces, so P(ace) = 4/52 = 1/13.
$1 - \frac{1}{13} = \frac{12}{13}$

Answer: 12/13

Direct

A die is thrown once. What is the probability of not getting a 6?

$P (6) = 1/6.$
$Q = 1 - \frac{1}{6} = \frac{5}{6}$

Answer: 5/6

Reverse

The probability that it will not rain tomorrow is 0.85. What is the probability that it will rain?

Try it first, then show the working

$P = 1 - 0.85 = 0.15$

Answer: 0.15

Exam

Two dice are thrown together. What is the probability that the sum is not 7?

Try it first, then show the working

There are 6 × 6 = 36 equally likely outcomes, and 6 of them add to 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
P(sum 7) = 6/36 = 1/6.
$1 - \frac{1}{6} = \frac{5}{6}$

Answer: 5/6

Common mistake: Treating two events as opposites when they are not: "not winning" includes a draw if draws are possible, so P(lose) is not always 1 − P(win).

Class 11

Addition rule of probability

P (A \cup B) = P (A) + P (B) - P (A \cap B)

What each letter means
$P (A \cup B)$	the probability that A or B (or both) happens (from 0 to 1)
$P (A)$	the probability of the event A (from 0 to 1)
$P (B)$	the probability of the event B (from 0 to 1)
$P (A \cap B)$	the probability that A and B both happen (from 0 to 1)

The probability that A or B happens is P(A) + P(B), less the probability that both happen, which was counted twice.

Why it works

Adding P(A) and P(B) counts every outcome in A and every outcome in B. The outcomes in both A and B are counted twice, so their probability P(A ∩ B) is taken away once.

When to use it

The probability of "A or B", "at least one of the two", and finding P(A ∩ B) when the others are known.

How to use it

Find P(A), P(B) and P(A ∩ B).
Add the first two and take away the third.
If A and B cannot happen together, P(A ∩ B) = 0 and the rule is plain adding.

To remember: Add both, take away the overlap.

Other forms

Special case $P (A \cup B) = P (A) + P (B)$ When A and B are mutually exclusive (they cannot happen together).
To remember itIt is the same rule as for counting a union: n(A ∪ B) = n(A) + n(B) − n(A ∩ B), divided through by the number of outcomes.

Worked examples

Story

The probability of rain on Monday is 0.3, on Tuesday 0.4, and on both days 0.1. Find the probability of rain on at least one of the two days.

$0.3 + 0.4 - 0.1 = 0.6$

Answer: 0.6

Picture

A dart hits a board. It lands in the red region with probability 0.35, in the inner circle with probability 0.25, and in the red part of the inner circle with probability 0.1. Find the probability that it lands in the red region or the inner circle.

$0.35 + 0.25 - 0.1 = 0.5$

Answer: 0.5

Direct

P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Find P(A ∪ B).

$0.5 + 0.4 - 0.2 = 0.7$

Answer: 0.7

Reverse

P(A ∪ B) = 0.8, P(A) = 0.5 and P(B) = 0.6. Find P(A ∩ B).

Try it first, then show the working

$0.5 + 0.6 - 0.8 = 0.3$

Answer: 0.3

Exam

One card is drawn from a pack of 52. Find the probability that it is a king or a heart.

Try it first, then show the working

P(king) = 4/52 and P(heart) = 13/52, and one card, the king of hearts, is both.
$\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{4}{13}$

Answer: 4/13

Common mistake: Adding P(A) and P(B) when the events overlap, which can even give a probability above 1.