Probability
1.1 Counting
↑ Back to TopStudents should be able to:
- count the arrangements of n distinct objects \((n!)\)
- count the number of ways of arranging r objects from n distinct objects
Key Concept: Factorial \((n!)\)
When all objects are distinct, the number of ways to arrange them in a line is \[ n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1. \]
For example, \(4! =4 \times 3 \times 2 \times 1 = 24\), so four different objects can be arranged in 24 ways.
Worked Example 1 - Factorial
Question:
How many different arrangements can be made using
the four letters A, B, C, D?
Solution:
There are 4 distinct letters, so the number of permutations is \[ 4! = 4 \times 3 \times 2 \times 1 = 24. \]
Answer: 24 arrangements.
Worked Example 2 — Permutation
Question:
A class has 8 different students taking part in a maths challenge.
Three students will be chosen and placed in 1st, 2nd, and 3rd position.
How many different possible podium orders are there?
Using the calculator (nPr)
We are arranging 3 from 8, so we use: \[ {}^{8}P_{3}. \]
On the calculator: \[ 8 \; \text{nPr} \; 3 = 336. \]
Alternative method — place-card idea
1st place: 8 choices
2nd place: 7 choices
3rd place: 6 choices
\[ 8 \times 7 \times 6 = 336. \]
Answer: 336 possible podium orders.
Arrangements - Quick Quiz
1.2 Concepts of Probability
↑ Back to TopStudents should be able to:
- use set theory to discuss experiments, outcomes, sample spaces
- apply AND/OR and mutually–exclusive rules using Venn diagrams
- calculate expected value and understand it need not be an outcome
- recognise expected value in decision making and fair games
Key ideas: experiments, outcomes, sample spaces
- Experiment: a process with an uncertain result (e.g. roll a die, spin a spinner).
- Outcome: a single possible result of the experiment (e.g. getting a 4).
- Sample space \(S\): the set of all possible outcomes.
Example: Toss two coins.
Outcomes: HH, HT, TH, TT
Sample space:
\[
S = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}.
\]
Each simple outcome has probability \(\dfrac14\) if the coins are fair.
Using Venn diagrams for AND / OR
In a class of 30 students:
- 18 study Irish (I)
- 14 study French (F)
- 8 study both subjects
Step 1: Fill the Venn diagram
Overlap (both): \(\#(I \cap F) = 8\)
Irish only: \(18 - 8 = 10\)
French only: \(14 - 8 = 6\)
Step 2: Use AND / OR
\(I \cap F\) (AND): \[ \#(I \cap F) = 8. \]
\(I \cup F\) (OR): students in Irish or French (or both) \[ \#(I \cup F) = 10 + 8 + 6 = 24. \] So, \[ P(I \cup F) = \frac{24}{30} = 0.8. \]
Mutually exclusive events
Two events are mutually exclusive if they cannot happen at the same time.
For mutually exclusive events \(A\) and \(B\): \[ \#(A \cap B) = 0 \] and \[ P(A \cup B) = P(A) + P(B). \]
Example:
Rolling a die — the events
“even number” and “odd number” are mutually exclusive.
Expected Value — A Fair Game
The expected value of a game tells you the average amount you would win if you played the game many, many times.
To find the expected value:
Multiply each outcome by its probability, then add the results.
Example: A fair spinner with the following outcomes:
| Outcome (€) | -2 | +1 | +3 |
|---|---|---|---|
| Probability | 1/3 | 1/3 | 1/3 |
Multiply each outcome by its probability:
\[ (-2)\left(\frac{1}{3}\right) + (1)\left(\frac{1}{3}\right) + 3\left(\frac{1}{3}\right) = \frac{-2 + 1 + 3}{3} = 0 \]
The expected value is €0.
Since the expected value is zero, this is a fair game.
Spinner Game – Win or Lose Money?
The spinner pays: Win €1 (50%), Lose €2 (30%), Win €3 (20%).
Answer the questions about this game. Click an option to check your answer.
1.3 Outcomes of Random Processes
↑ Back to TopStudents should be able to:
- find the probability that two independent events occur together
- apply understanding of Bernoulli trials
- solve problems involving up to three Bernoulli trials
- calculate the probability that the first success occurs on the \(n^{\text{th}}\) trial
Tree Diagrams – Red and Blue Counters
A counter is randomly selected from a bag containing 4 red counters and 6 blue counters. It is then replaced and a second counter is selected.
Use the tree diagram to answer the quiz questions below.
Check Your Understanding
Question 1 of 5Bernoulli Trials – “First Success on the nth Trial”
In these questions we repeat the same experiment many times (e.g., rolling a die). Because the probability stays the same each time, this is a Bernoulli Trial.
We answer questions such as:
- “The probability that the first 6 appears on the 3rd throw.”
- “The probability that the first even number appears on the 2nd throw.”
To have the first success on the \(n\)th trial, the sequence must be:
fail, fail, …, fail, success.
Steps:
- Identify the probabilities of success and failure (success = 6 → \(P(6)=\frac16\), \(P(\text{not 6})=\frac56\)).
- Write the sequence: not 6, not 6, 6.
- Multiply the probabilities.
First 6 on the 3rd throw:
\(P(\text{not 6, not 6, 6})\)
\(= \left(\frac56\right)\left(\frac56\right)\left(\frac16\right)\)
\(= \frac{25}{216} \approx 0.1157.\)
Bernoulli Trials – First Success Quiz
A bag contains 7 yellow counters and 3 green counters. One counter is selected at random, its colour is noted and it is replaced. This process is repeated independently.
For each question, think about a sequence of independent trials where the same probability is used each time.
Check Your Understanding
Question 1 of 31.1 Counting
Students should be able to:
- count the arrangements of \(n\) distinct objects \((n!)\)
- count the number of ways of arranging \(r\) objects from \(n\) distinct objects
1.2 Concepts of probability
Students should be able to:
- use set theory to discuss experiments, outcomes, sample spaces
- apply AND/OR and mutually–exclusive rules using Venn diagrams
- calculate expected value and understand it need not be an outcome
- recognise expected value in decision making and fair games
1.3 Outcomes of random processes
Students should be able to:
- find the probability that two independent events occur together
- apply understanding of Bernoulli trials
- solve problems involving up to three Bernoulli trials
- calculate the probability that the first success occurs on the \(n^\text{th}\) trial
