SPI Mathematics · Probability

Coins and a Single Die

The probability of an event is the number of outcomes you want divided by the number of equally likely outcomes there are.

All Probability notes

Why this matters

Twelve questions in the bank, all EASY: a die thrown once, or two or three coins tossed. Each is one count over 6, 4 or 8, so the only way to lose the mark is to list the outcomes wrongly.

Concept 1 of 4: Experiment, outcome, sample space and event

Throwing a die is an experiment. Each face it can show is an outcome, and the list of all outcomes is the sample space. An event is any group of outcomes you care about, such as 'an even number'.

Definition

Sample space for one die: $\{1, 2, 3, 4, 5, 6\}$ , six outcomes.
Sample space for one coin: $\{H, T\}$ , two outcomes.
An event is a part of the sample space. 'An even number' on a die is $\{2, 4, 6\}$ .
Outcomes are equally likely when none is favoured: a fair die, a fair coin, a well-shuffled deck.

Number of outcomes

n(S) = \text{number of outcomes in the sample space}

Worked example

List the sample space for one die and the event 'a number greater than 4'.

Concept 2 of 4: Probability as favourable outcomes over all outcomes

When every outcome is equally likely, the chance of an event is simply the share of outcomes that belong to it. Count the ones you want, count all of them, divide.

Definition

$P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .
Simplify the fraction at the end: $\frac{3}{6} = \frac{1}{2}$ .
The rule needs equally likely outcomes. A die with letters or colours on its faces still has six equally likely faces.

Classical probability

P(E) = \frac{n(E)}{n(S)}

n(E)number of outcomes favourable to the event
n(S)number of outcomes in the sample space

Worked example

A die is thrown once. Find the probability of a number greater than 4.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Probability · Coins and a Single Die

A dice is throw, find the probability of getting not even number?

Forgetting to simplify

\frac{3}{6}

is usually printed as

\frac{1}{2}

. If your fraction is not among the options, reduce it before deciding you are wrong.

'Not even' is not a new count

'Not an even number' on a die means 1, 3 or 5: the odd numbers. Rephrase a 'not' into the outcomes it leaves before counting.

Drill 2 more on probability as favourable outcomes over all outcomes

Concept 3 of 4: Which faces of a die are prime, composite or square

Most single-die questions are really number questions in disguise: 'a prime number', 'a perfect square', 'a multiple of 3'. Know which of 1 to 6 fit each word and the probability is immediate.

Definition

Two facts settle most disputes:

1 is neither prime nor composite.
2 is the only even prime, so the odd primes on a die are 3 and 5.

Property	Faces	Probability
Even	2, 4, 6	1/2
Odd	1, 3, 5	1/2
Prime	2, 3, 5	1/2
Odd prime	3, 5	1/3
Composite	4, 6	1/3
Perfect square	1, 4	1/3
Multiple of 3	3, 6	1/3
Neither prime nor composite	1	1/6 1 has only one factor, so it is in neither list.

Every probability is the number of faces over 6, reduced.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Probability · Coins and a Single Die

A die is thrown. The probability of getting an odd prime number is ____.

Counting 1 as a prime

1 is not prime, so the primes on a die are 2, 3 and 5, giving

\frac{1}{2}

, not

\frac{2}{3}

Counting 2 as an odd prime

2 is prime but even. 'An odd prime' on a die is 3 or 5 only.

Drill 3 more on which faces of a die are prime, composite or square

Concept 4 of 4: Listing the outcomes for two or three coins

Each coin doubles the list: one coin has 2 outcomes, two coins 4, three coins 8. Write them all out in a fixed order and count the ones that fit.

Definition

Two coins: HH, HT, TH, TT. HT and TH are different outcomes.
Three coins: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
Exactly one head, at least one head and at most one head are three different events.
The same doubling works for dice: each die multiplies the count by 6, so three dice give $6^3 = 216$ outcomes.

Outcomes for k coins or dice

n(S) = 2^k \text{ for } k \text{ coins} \qquad n(S) = 6^k \text{ for } k \text{ dice}

Worked example

Three coins are tossed. Find the probability of exactly two heads.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Probability · Coins and a Single Die

If three coins are tossed simultaneously, what is the probability of getting at most one tail?

Treating HT and TH as one outcome

With two coins there are four outcomes, not three. Exactly one head is HT or TH, so its probability is

\frac{2}{4} = \frac{1}{2}

, not

\frac{1}{3}

'At most one' includes none

At most one tail with three coins means no tail or one tail: HHH, HHT, HTH, THH. Leaving out HHH gives

\frac{3}{8}

, a printed wrong option.

Drill 4 more on listing the outcomes for two or three coins

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Experiment, outcome, sample space and event
Number of outcomes
$n(S) = \text{number of outcomes in the sample space}$
Probability as favourable outcomes over all outcomes
Classical probability
$P(E) = \frac{n(E)}{n(S)}$
Listing the outcomes for two or three coins
Outcomes for k coins or dice
$n(S) = 2^k \text{ for } k \text{ coins} \qquad n(S) = 6^k \text{ for } k \text{ dice}$

Reference tables (1)

Which faces of a die are prime, composite or square8 rows

Property	Faces	Probability
Even	2, 4, 6	1/2
Odd	1, 3, 5	1/2
Prime	2, 3, 5	1/2
Odd prime	3, 5	1/3
Composite	4, 6	1/3
Perfect square	1, 4	1/3
Multiple of 3	3, 6	1/3
Neither prime nor composite	1	1/6 1 has only one factor, so it is in neither list.

Every probability is the number of faces over 6, reduced.

Watch out for (6)

Forgetting to simplify
→ Probability as favourable outcomes over all outcomes
'Not even' is not a new count
→ Probability as favourable outcomes over all outcomes
Counting 1 as a prime
→ Which faces of a die are prime, composite or square
Counting 2 as an odd prime
→ Which faces of a die are prime, composite or square
Treating HT and TH as one outcome
→ Listing the outcomes for two or three coins
'At most one' includes none
→ Listing the outcomes for two or three coins

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