Need to Know

Probability

The rules you must have ready for the exam.

How to use this: say each rule in your head, then tap Show me to check.

Basic probability

3 rules

Rule 01

Probability is a fraction.

$P(\text{event}) = \dfrac{\text{number of ways it can happen}}{\text{total number of outcomes}}$.

Rule 02

The scale runs 0 to 1.

$0$ = impossible, $\tfrac{1}{2}$ = even ($50:50$), $1$ = certain. A probability is never below $0$ or above $1$.

Rule 03

The complement.

$P(\text{not } A) = 1 - P(A)$. The chance something does not happen is $1$ minus the chance it does.

Counting & two events

4 rules

Rule 04

AND means multiply, OR means add.

For one thing and another, multiply the ways. For one or another, add.

Rule 05

Multiply along a tree.

For two draws, multiply along the branches for each outcome and add the outcomes you want.

Rule 06

With vs without replacement.

Replaced: the fractions stay the same. Not replaced: the total drops by $1$ and so does the colour taken.

Rule 07

At least one.

$P(\text{at least one}) = 1 - P(\text{none})$. Easier to take away the ‘none’ case.

Venn diagrams & ‘or’

3 rules

Rule 08

Reading a Venn.

The box is everything; the overlap is both. Probability = the region you want $\div$ the total in the box.

Rule 09

The ‘or’ rule.

$P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$ — take away the overlap so it isn’t counted twice.

Rule 10

Mutually exclusive.

Events with nothing in the overlap. Then $P(A \text{ or } B) = P(A) + P(B)$.

Expected value

1 rule

Rule 11

Average per play.

$E(x) = \sum x \cdot P(x)$ — each prize times its probability, all added. Subtract the cost to see if the game is worth playing.

Permutations

4 rules

Rule 12

Fill boxes and multiply.

Draw a box for each position, write the choices for each, and multiply. Each choice used leaves one fewer.

Rule 13

Factorial.

$n!$ = the number of ways to arrange all of $n$ different things: $n! = n \times (n-1) \times \cdots \times 1$.

Rule 14

Fill the fussy box first.

If a position has a rule (starts with a certain letter, must be odd…), fill that box first, then the rest.

Rule 15

Keep items together.

Bubble the items as one, arrange the bubbles, then multiply by the ways inside the bubble.

Mathslive.ie · Probability · Need to Know