Probability · Class 10 Mathematics: quick revision, 34 bites

What probability is

Counting, carefully

Takeaway. Wanted over total.

Equally likely

The hidden condition

The definition only works when no outcome is more likely than another.

Takeaway. Always ask whether the outcomes are balanced.

The safest habit

List before you count

Two coins give HH, HT, TH, TT — FOUR outcomes, not three.

Takeaway. Listing catches the classic error.

Two different words

Outcome against event

An OUTCOME is one result; an EVENT is a collection of them.

Takeaway. Count the outcomes in the event.

0 to 1

Never outside the range

Takeaway. An answer above 1 is an arithmetic error, not a surprise.

Impossible and certain

The two extremes

$P(E) = 0$: no outcome is favourable, so the event cannot happen.

Takeaway. Rolling a 7 on one die has probability 0.

The complement

Everything else

Takeaway. The event either happens or it does not.

Using the complement

Not a king

Takeaway. Far quicker than counting 48 cards.

Harder than its opposite

When to reach for it

'Not', 'at least' and 'none' usually mean the complement is shorter.

Takeaway. Ask the question before you start counting.

Two dice

Not a seven

Takeaway. One short count instead of one long one.

A free check

They all add to one

The probabilities of all the outcomes of an experiment sum to exactly 1.

Takeaway. If a list does not, something is missing or wrong.

How to write it

Lowest terms

Give $\frac12$, not $\frac{18}{36}$.

Takeaway. A fraction, a decimal or a percentage — not odds.

Reading the words

At least is not exactly

'At least one head' includes two heads; 'exactly one head' does not.

Takeaway. Underline the phrase before counting.

Every question

The five steps

Takeaway. Step 4 long? Try the complement.

And nothing less

Zero means impossible

A very small probability is not zero — only an event that cannot occur at all has probability 0.

Takeaway. Unlikely and impossible are different.

A seventeenth-century question

It began with gamblers

Two players asked how to split the stakes of a game abandoned half-way through.

Takeaway. The answer needed probability, and so it was invented.

70% rain

Percentages are probabilities

A forecast of 70% rain is the probability $\frac{7}{10}$ in another dress.

Takeaway. Fraction, decimal, percentage — the same number.

What to memorise

Three experiments, endless questions

Nearly every question is about coins, dice or a pack of cards.

Takeaway. Memorise once, use everywhere.

Counting coin outcomes

Each coin doubles it

Takeaway. 2, 4 and 8 for one, two and three coins.

The eight outcomes

Three coins listed

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Takeaway. Exactly two heads happens three ways.

A subtle trap

Head counts are not equal

Three coins give four possible head-counts — 0, 1, 2, 3 — but they are NOT equally likely.

Takeaway. Count outcomes, never categories.

Two dice

Thirty-six, not twenty-one

$(2,5)$ and $(5,2)$ are DIFFERENT outcomes, even if the dice look alike.

Takeaway. Treating them as one wrecks every later answer.

Two dice

The shape of the totals

Ways to make each total: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 for totals 2 to 12.

Takeaway. Seven is the commonest, 2 and 12 the rarest.

Doubles

Both the same

$(1,1)$ through $(6,6)$ — six outcomes.

Takeaway. The same as a total of seven, by coincidence.

The pack

Four thirteens

52 cards: spades and clubs black, hearts and diamonds red.

Takeaway. Learn this once and never recount it.

The chapter's favourite trap

An ace is not a face card

FACE cards are the jack, queen and king only: 3 per suit, 12 in the pack.

Takeaway. Say it out loud before counting.

Suit and rank

Two to have by heart

Takeaway. A heart is a quarter; a king is a thirteenth.

Red kings and face cards

Combining them

Takeaway. Count the cards, then divide by 52.

Use the complement

At least one

With three coins, only TTT has no head.

Takeaway. One count instead of seven.

A king OR a heart

Overlapping descriptions

4 kings plus 13 hearts gives 17, but the king of hearts is in both lists.

Takeaway. List the cards rather than adding the counts.

When cards are removed

The total changes too

Remove the 12 face cards and 40 remain, with only 10 hearts left.

Takeaway. Adjust the denominator as well as the numerator.

Where marks are lost

Write the total first

Write down the total number of outcomes before anything else, adjusted for removals.

Takeaway. It is the commonest error in the chapter.

Exactly, at least, or

Read the English

The arithmetic here is easy; the wording is not.

Takeaway. Most errors are reading errors.

The commonest total

Why games love seven

Seven can be made six ways out of thirty-six, more than any other total.

Takeaway. Which is why so many dice games turn on it.