Events and Outcomes Revision Notes for Leaving Cert Mathematics

Events and Outcomes

Key definitions

Understanding probability starts with three essential terms that form the foundation of all probability work.

A trial is the action or process we carry out to generate a result. When you throw a dice to start a board game, the act of throwing that dice is the trial. The trial is what we do, not what we get.

An outcome refers to any possible result that can occur from a trial. When throwing a standard dice, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. These represent every single result that could happen.

An event is the specific result or group of results that we are interested in from our trial. If you need to roll an even number to start a game, then getting 2, 4, or 6 would be your event. The event is what we want to happen.

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These three terms work together in every probability situation. First you perform a trial, which can produce various outcomes, and you're usually interested in the probability of a specific event occurring.

The fundamental probability formula

Probability measures how likely an event is to occur and is calculated using this essential formula:

$P(E) = \frac{\text{number of successful outcomes in E}}{\text{number of possible outcomes}}$

This formula tells us that probability is simply a fraction. The top of the fraction (numerator) counts how many ways our desired event can happen. The bottom (denominator) counts all the possible outcomes that could occur.

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This formula only works when all outcomes are equally likely to occur. Always check that this condition is met before applying the formula.

Essential probability rules

There are three crucial rules that govern all probability calculations:

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The Three Fundamental Rules of Probability:

Probability range: The probability of any event E must be between 0 and 1 inclusive, written as $0 \leq P(E) \leq 1$ . This means probability is always a fraction, decimal, or percentage between these values.
Certainty: When something is guaranteed to happen, its probability equals 1. If you're certain to get a number between 1 and 6 when rolling a fair dice, then $P(\text{number between 1 and 6}) = 1$ .
Impossibility: When something cannot possibly happen, its probability equals 0. Rolling a 7 on a standard dice has probability $P(\text{rolling 7}) = 0$ .

Equally likely outcomes

Two or more events are equally likely when they have exactly the same chance of happening. This concept is fundamental to fair probability situations.

Consider a spinner divided into sections. If the red and blue sections are the same size, then landing on red and landing on blue are equally likely events. However, if the sections are different sizes, the events are not equally likely.

For a pentagon spinner with five equal sections, each section has the same probability of being selected. We can calculate specific probabilities:

$P(\text{green}) = \frac{1}{5}$ because 1 out of 5 sections is green
$P(\text{yellow}) = \frac{2}{5}$ because 2 out of 5 sections are yellow

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When outcomes are equally likely, you can simply count the favourable outcomes and divide by the total number of outcomes. This makes calculations much easier!

Important probability terminology

In probability questions, certain words have special meanings that indicate all outcomes are equally likely:

Random: When something is selected at random, each possible choice has an equal chance of being picked
Fair: A fair dice, coin, or spinner means all outcomes are equally likely to occur

These terms are crucial for recognising when to apply the basic probability formula. They tell you that you can count outcomes and use the simple fraction approach.

Worked examples

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Worked Example: Drawing numbered tickets

Tickets numbered 1 to 12 are placed in a box. If one ticket is drawn at random, find the probability of getting: (i) the number 4 (ii) an even number
(iii) a two-digit number (iv) a number divisible by 4

Solutions:

(i) There is one ticket with 4 out of 12 total tickets $P(4) = \frac{1}{12}$

(ii) Even numbers from 1 to 12 are: 2, 4, 6, 8, 10, 12 (6 numbers) $P(\text{even number}) = \frac{6}{12} = \frac{1}{2}$

(iii) Two-digit numbers from 1 to 12 are: 10, 11, 12 (3 numbers) $P(\text{two-digit number}) = \frac{3}{12} = \frac{1}{4}$

(iv) Numbers divisible by 4 from 1 to 12 are: 4, 8, 12 (3 numbers) $P(\text{divisible by 4}) = \frac{3}{12} = \frac{1}{4}$

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Worked Example: Drawing from a pack of cards

If a card is drawn from a standard pack of 52 cards, find the probability that it is: (i) a king (ii) a spade (iii) a red card

Solutions:

(i) There are 4 kings in a pack of 52 cards $P(\text{king}) = \frac{4}{52} = \frac{1}{13}$

(ii) There are 13 spades in a pack of 52 cards $P(\text{spade}) = \frac{13}{52} = \frac{1}{4}$

(iii) There are 26 red cards (hearts and diamonds) in a pack of 52 $P(\text{red card}) = \frac{26}{52} = \frac{1}{2}$

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Worked Example: Selecting letters from a word

A letter is selected at random from the letters of the word COMPANION. Find the probability that the letter is: (i) P (ii) N (iii) a vowel (iv) M or N

Solutions:

The word COMPANION has 9 letters: C-O-M-P-A-N-I-O-N

(i) There is 1 letter P in 9 letters $P(P) = \frac{1}{9}$

(ii) There are 2 letter Ns in 9 letters
$P(N) = \frac{2}{9}$

(iii) The vowels are O, A, I, O (4 vowels total) $P(\text{vowel}) = \frac{4}{9}$

(iv) There is 1 M and 2 Ns, making 3 letters total $P(M \text{ or } N) = \frac{3}{9} = \frac{1}{3}$

Exam tips and common mistakes

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Common Pitfalls to Avoid:

Always identify the total number of possible outcomes first
Count carefully - make lists if necessary to avoid missing outcomes
Simplify fractions to their lowest terms
Remember that "or" in probability usually means addition of favourable outcomes
Check that your probability answer is between 0 and 1
When dealing with cards, remember: 52 cards total, 4 suits of 13 cards each, 26 red and 26 black cards

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Key Points to Remember:

A trial is the action performed, outcomes are all possible results, and an event is the specific result you want
Probability formula: $P(E) = \frac{\text{successful outcomes}}{\text{total outcomes}}$
All probabilities must be between 0 and 1 inclusive
Equally likely outcomes have the same chance of occurring
Words like random and fair indicate all outcomes are equally likely

Events and Outcomes (Leaving Cert Mathematics): Revision Notes