Section A — Chapters 1–22

The Fundamental Principle of Counting and Probability

Junior Cycle — 1st Year

✓List all possible outcomes for simple events.
✓Apply the Fundamental Principle of Counting to determine the total number of outcomes.
✓Understand and calculate the theoretical probability of an event.
✓Distinguish between relative frequency (experimental probability) and theoretical probability.
✓Calculate the expected frequency of an event and construct sample spaces for successive and independent events.

Key concepts

Listing Outcomes

Listing outcomes involves systematically writing down all the possible results that can occur from an experiment or event. This ensures no outcome is missed.

Fundamental Principle of Counting

If there are 'm' ways for one event to occur and 'n' ways for a second independent event to occur, then there are 'm x n' ways for both events to occur. This principle can be extended to more than two events.

Introducing Probability

Probability is a measure of how likely an event is to happen. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

Relative Frequency (Experimental Probability)

Relative frequency, also known as experimental probability, is calculated from the results of an actual experiment. It is the number of times a specific event occurs divided by the total number of trials in the experiment.

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

Relative Frequency & Fairness

For a fair experiment, as the number of trials increases, the relative frequency of an event tends to get closer to its theoretical probability. If the relative frequency consistently deviates significantly from the theoretical probability over many trials, it may suggest the experiment is not fair.

Theoretical Probability

Theoretical probability is based on reasoning and calculation, assuming all possible outcomes are equally likely. It represents what 'should' happen in an ideal situation.

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

Expected Frequency

Expected frequency is the predicted number of times an event is likely to occur in a given number of trials, based on its theoretical probability.

Expected Frequency = P(Event) x (Number of trials)

Sample Spaces for Successive & Independent Events

A sample space is a list or set of all possible outcomes of an experiment. For successive events (events that happen one after another) or independent events (where the outcome of one does not affect the outcome of the other), the sample space can be constructed using lists, tables, or tree diagrams.

Key facts to remember

1Probability values always lie between 0 (impossible) and 1 (certain), inclusive.
2The Fundamental Principle of Counting states that if there are 'm' ways for one event and 'n' ways for another, there are 'm x n' ways for both.
3Theoretical Probability = (Number of favourable outcomes) / (Total number of possible outcomes).
4Relative Frequency (Experimental Probability) = (Number of times an event occurs) / (Total number of trials).
5Expected Frequency = P(Event) x (Number of trials).
6A sample space is a list of all possible outcomes of an experiment.
7The sum of the probabilities of all possible outcomes in a sample space is 1.
8For a fair experiment, relative frequency tends towards theoretical probability as the number of trials increases.

Worked examples

Example 1

A restaurant offers 4 main courses (beef, chicken, fish, vegetarian) and 3 desserts (ice cream, cake, fruit). How many different two-course meal combinations are possible? List all the possible combinations.

IIdentify the number of choices for each part of the meal: Main courses = 4, Desserts = 3.

IIApply the Fundamental Principle of Counting: Total combinations = (Number of main courses) x (Number of desserts).

IIICalculate the total number of combinations: 4 x 3 = 12.

IVList all possible combinations systematically:

VBeef & Ice Cream, Beef & Cake, Beef & Fruit

VIChicken & Ice Cream, Chicken & Cake, Chicken & Fruit

VIIFish & Ice Cream, Fish & Cake, Fish & Fruit

VIIIVegetarian & Ice Cream, Vegetarian & Cake, Vegetarian & Fruit

Answer

There are 12 different two-course meal combinations possible. The combinations are: (Beef, Ice Cream), (Beef, Cake), (Beef, Fruit), (Chicken, Ice Cream), (Chicken, Cake), (Chicken, Fruit), (Fish, Ice Cream), (Fish, Cake), (Fish, Fruit), (Vegetarian, Ice Cream), (Vegetarian, Cake), (Vegetarian, Fruit).

Example 2

A fair spinner has 5 equally sized sections coloured Red, Blue, Green, Yellow, and Purple. It is spun 100 times.

I(a) Calculate the theoretical probability of landing on 'Blue'.

IIP(Blue) = (Number of blue sections) / (Total number of sections) = 1/5.

III(b) How many times would you expect the spinner to land on 'Blue' in 100 spins?

IVExpected Frequency = P(Blue) x (Number of trials) = (1/5) x 100 = 20.

V(c) If the spinner actually landed on 'Blue' 18 times in 100 spins, what is the relative frequency of landing on 'Blue'?

VIRelative Frequency = (Number of times 'Blue' occurred) / (Total number of trials) = 18/100 = 9/50.

VII(d) Based on the relative frequency, do you think the spinner is fair? Explain.

VIIICompare the relative frequency (9/50 = 0.18) to the theoretical probability (1/5 = 0.2). The relative frequency is close to the theoretical probability. For 100 trials, a slight variation is expected. Therefore, it is likely that the spinner is fair.

Answer

(a) P(Blue) = 1/5.\n(b) You would expect the spinner to land on 'Blue' 20 times.\n(c) The relative frequency of landing on 'Blue' is 18/100 or 9/50.\n(d) Yes, the spinner appears fair. The relative frequency (0.18) is close to the theoretical probability (0.2), which is expected for a limited number of trials.

Remember that relative frequency approaches theoretical probability as the number of trials increases. Small differences are normal in experimental results.

Example 3

A bag contains 2 red (R) marbles and 1 blue (B) marble. A marble is drawn, its colour is noted, and it is replaced. Then a second marble is drawn and its colour is noted.

I(a) Construct the sample space for this experiment.

IIFor the first draw, the possible outcomes are R1, R2, B. Since the marbles are replaced, the second draw also has outcomes R1, R2, B. We can use a tree diagram or a table.

IIILet's list them: (First Draw, Second Draw)

IVR1R1, R1R2, R1B

VR2R1, R2R2, R2B

VIBR1, BR2, BB

VIITo simplify, since the red marbles are identical in colour, we can represent the outcomes as (Red, Red), (Red, Blue), (Blue, Red), (Blue, Blue).

VIIITotal possible outcomes = 3 x 3 = 9 if we distinguish between individual red marbles (R1, R2). If we only care about the colour 'Red', then the outcomes are (R,R), (R,B), (B,R), (B,B). Let's use the distinct marble approach for clarity in sample space, then simplify for probability.

9Sample Space: { (R1,R1), (R1,R2), (R1,B), (R2,R1), (R2,R2), (R2,B), (B,R1), (B,R2), (B,B) }

10Total outcomes = 9.

11(b) What is the probability of drawing two red marbles?

12Favourable outcomes for two red marbles are (R1,R1), (R1,R2), (R2,R1), (R2,R2). There are 4 such outcomes.

13P(Two Red Marbles) = (Number of favourable outcomes) / (Total number of outcomes) = 4/9.

Answer

(a) The sample space is: { (R1,R1), (R1,R2), (R1,B), (R2,R1), (R2,R2), (R2,B), (B,R1), (B,R2), (B,B) }.\n(b) The probability of drawing two red marbles is 4/9.

When constructing a sample space, ensure all individual outcomes are distinct. If marbles of the same colour are indistinguishable for the final probability, you can group them after listing the distinct outcomes.

Common mistakes

✗Confusing theoretical probability with relative frequency, especially when interpreting experimental results.
✗Failing to list all possible outcomes systematically when constructing a sample space, leading to an incorrect total number of outcomes.
✗Incorrectly applying the Fundamental Principle of Counting, often by adding instead of multiplying.
✗Expressing probability as a number less than 0 or greater than 1.
✗Not simplifying probability fractions to their lowest terms.

Exam tips

★Always show your full working for probability questions, including the formula used and the substitution of values.
★Use tree diagrams or tables to help visualise and construct sample spaces for experiments with multiple stages.
★Read the question carefully to determine whether theoretical probability, relative frequency, or expected frequency is being asked for.
★Ensure your final probability answers are given as simplified fractions, decimals, or percentages, unless otherwise specified in the question.

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