Strand 1 — Statistics & Probability

Probability

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to define and identify outcomes, events, and sample space.
✓By the end of this lesson students will be able to calculate the theoretical probability of an event.
✓By the end of this lesson students will be able to understand and calculate relative frequency (experimental probability).
✓By the end of this lesson students will be able to compare relative frequency with theoretical probability.
✓By the end of this lesson students will be able to calculate probabilities of two independent events, including using tree diagrams.

Key concepts

Outcomes and Events

An 'outcome' is a possible result of an experiment or trial. The 'sample space' (S) is the set of all possible outcomes. An 'event' (E) is a specific collection of one or more outcomes from the sample space. For example, when rolling a standard six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}. An event could be 'rolling an even number', so E = {2, 4, 6}.

Theoretical Probability

Theoretical probability is the likelihood of an event occurring based on reasoning, assuming all outcomes are equally likely. It is calculated by dividing the number of favourable outcomes for an event by the total number of possible outcomes in the sample space.

P(E) = (Number of favourable outcomes) / (Total number of possible outcomes)

Relative Frequency (Experimental Probability)

Relative frequency, also known as experimental probability, is the probability of an event based on actual experiments or observations. It is calculated by dividing the number of times an event occurs during an experiment by the total number of trials performed.

Relative Frequency = (Number of times an event occurs) / (Total number of trials)

Comparing Relative Frequency and Theoretical Probability

While theoretical probability tells us what we expect to happen in an ideal situation, relative frequency tells us what actually happened in an experiment. The Law of Large Numbers states that as the number of trials in an experiment increases, the relative frequency of an event tends to get closer to its theoretical probability.

Two-Event Probability

Two-event probability involves calculating the likelihood of two events happening. If two events, A and B, are 'independent', meaning the outcome of one does not affect the outcome of the other, the probability of both events occurring is found by multiplying their individual probabilities. If two events are 'mutually exclusive', meaning they cannot both happen at the same time, the probability of either event A or event B occurring is the sum of their individual probabilities.

For independent events A and B: P(A and B) = P(A) * P(B) For mutually exclusive events A and B: P(A or B) = P(A) + P(B)

Tree Diagrams

A tree diagram is a visual tool used to list all possible outcomes of a sequence of events and to calculate their probabilities. Each branch of the 'tree' represents a possible outcome, and the probability of that outcome is written along the branch. To find the probability of a sequence of outcomes, you multiply the probabilities along the path of the branches that lead to that specific sequence.

Key facts to remember

1Probability values always range from 0 (impossible event) to 1 (certain event), inclusive.
2The sum of the probabilities of all possible outcomes in a sample space is always 1.
3P(E) = (Number of favourable outcomes) / (Total number of possible outcomes) for theoretical probability.
4Relative frequency is calculated from experimental results: (Number of times event occurs) / (Total number of trials).
5For two independent events A and B, the probability of both occurring is P(A and B) = P(A) * P(B).
6A tree diagram is a useful visual tool for listing all possible outcomes and calculating probabilities for sequential events.
7The Law of Large Numbers states that as the number of trials increases, relative frequency tends to approach theoretical probability.

Worked examples

Example 1

A bag contains 4 red, 6 blue, and 2 green marbles. A marble is chosen at random from the bag. What is the probability that it is: (a) red? (b) blue or green? (c) not green?

IFirst, find the total number of possible outcomes (total marbles): 4 + 6 + 2 = 12 marbles.

II(a) For the event 'red marble': Number of favourable outcomes = 4 (red marbles). P(red) = (Number of red marbles) / (Total number of marbles) = 4/12.

III(b) For the event 'blue or green marble': Number of favourable outcomes = 6 (blue) + 2 (green) = 8 marbles. P(blue or green) = (Number of blue or green marbles) / (Total number of marbles) = 8/12.

IV(c) For the event 'not green marble': Number of favourable outcomes = 4 (red) + 6 (blue) = 10 marbles. Alternatively, P(not green) = 1 - P(green). P(green) = 2/12. P(not green) = 1 - 2/12 = 12/12 - 2/12 = 10/12.

Answer

(a) P(red) = 1/3 (b) P(blue or green) = 2/3 (c) P(not green) = 5/6

Always simplify your probability fractions to their lowest terms.

Example 2

A spinner has three equal sections labelled A, B, and C. It is spun 80 times. The results are: A occurs 35 times, B occurs 25 times, and C occurs 20 times. (a) What is the relative frequency of landing on A? (b) What is the theoretical probability of landing on A? (c) Explain why the answers to (a) and (b) might be different.

I(a) Relative frequency of landing on A = (Number of times A occurred) / (Total number of spins) = 35/80.

II(b) For a spinner with three equal sections, the total number of possible outcomes is 3 (A, B, C). The number of favourable outcomes for landing on A is 1. Theoretical P(A) = (Number of favourable outcomes) / (Total number of possible outcomes) = 1/3.

III(c) The relative frequency is based on an actual experiment with a limited number of trials (80 spins). The theoretical probability is what we expect in an ideal situation. The difference occurs because random events do not always perfectly match theoretical expectations over a small number of trials. If the spinner were spun many more times, the relative frequency would likely get closer to the theoretical probability of 1/3.

Answer

(a) Relative Frequency (A) = 7/16 (b) Theoretical P(A) = 1/3 (c) The relative frequency is based on experimental results, while theoretical probability is based on ideal conditions. With a limited number of trials, the experimental results may not perfectly match the theoretical expectation.

Example 3

A fair coin is tossed and a fair four-sided spinner (numbered 1, 2, 3, 4) is spun. (a) Draw a tree diagram to show all possible outcomes. (b) What is the probability of getting a Head and an odd number? (c) What is the probability of getting a Tail and a number greater than 2?

I(a) Draw the tree diagram: Coin Toss (1st event): - Head (H) with P(H) = 1/2 - Tail (T) with P(T) = 1/2 Spinner Spin (2nd event, from each coin outcome): - From H: 1 (P=1/4), 2 (P=1/4), 3 (P=1/4), 4 (P=1/4) - From T: 1 (P=1/4), 2 (P=1/4), 3 (P=1/4), 4 (P=1/4) Possible outcomes and their probabilities (multiply along branches): (H,1) P = 1/2 * 1/4 = 1/8 (H,2) P = 1/2 * 1/4 = 1/8 (H,3) P = 1/2 * 1/4 = 1/8 (H,4) P = 1/2 * 1/4 = 1/8 (T,1) P = 1/2 * 1/4 = 1/8 (T,2) P = 1/2 * 1/4 = 1/8 (T,3) P = 1/2 * 1/4 = 1/8 (T,4) P = 1/2 * 1/4 = 1/8

II(b) Event: Head and an odd number. The outcomes are (H,1) and (H,3). P(H and odd) = P(H,1) + P(H,3) = 1/8 + 1/8 = 2/8.

III(c) Event: Tail and a number greater than 2. The numbers greater than 2 are 3 and 4. The outcomes are (T,3) and (T,4). P(T and >2) = P(T,3) + P(T,4) = 1/8 + 1/8 = 2/8.

Answer

(a) Tree diagram showing outcomes (H,1), (H,2), (H,3), (H,4), (T,1), (T,2), (T,3), (T,4), each with a probability of 1/8. (b) P(Head and odd number) = 1/4 (c) P(Tail and number greater than 2) = 1/4

When using a tree diagram for independent events, multiply probabilities along the branches to find the probability of a specific sequence of outcomes.

Common mistakes

✗Not listing all possible outcomes correctly when determining the sample space.
✗Confusing 'and' (multiplication for independent events) with 'or' (addition for mutually exclusive events) in probability calculations.
✗Forgetting to simplify probability fractions to their lowest terms.
✗Assuming that relative frequency must be exactly equal to theoretical probability after only a small number of trials.
✗Incorrectly adding probabilities for events that are not mutually exclusive without subtracting the overlap.

Exam tips

★Always clearly define your sample space (S) and the event (E) you are interested in.
★Simplify all probability fractions to their lowest terms, unless otherwise specified.
★For problems involving two or more events, consider using a two-way table or a tree diagram to organise the information and ensure all outcomes are considered.
★Read the question carefully to determine if events are independent or dependent, and whether you need to calculate 'and' or 'or' probabilities.
★Always check that your final probability answer is a value between 0 and 1, inclusive. If it's outside this range, you've made a mistake.

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