SkillrHub • Year 7 Maths

AC9M7P01 • Homework

Sample Spaces, Probability and Relative Frequency

List outcomes carefully, state when outcomes are equally likely, distinguish theoretical probability from experimental relative frequency, and treat expected frequency as a prediction rather than a guarantee.

5 × 2 marks
Short response
5 × 5 marks
Extended reasoning
1 enrichment
Prediction application

Part A — Short response

  1. Define sample space and give the sample space for one fair coin toss. (2)
  2. List the sample space for rolling one fair six-sided die. (2)
  3. Find P(heads) for one fair coin toss. (2)
  4. Define a favourable outcome and give an example using a die. (2)
  5. Three equally likely outcomes are Red, Blue and Green. Find P(Red). (2)

Part B — Extended response

  1. A spinner has four equal sectors: Red, Blue, Green, Yellow. List the sample space, find P(Green), predict Green outcomes in 80 spins, and explain why the prediction need not occur exactly. (5)
  2. A fair die is rolled 60 times and 6 appears 8 times. Calculate the relative frequency, compare it with 1/6, and explain the difference. (5)
  3. A lucky dip has five equally likely distinct prizes A–E. List the sample space, find P(C), predict C in 100 draws with replacement, and state the assumption that makes 1/5 valid. (5)
  4. A fair coin is tossed 100 times and gives 47 heads. Find the relative frequency, compare it with 0.5, then give both the theoretical prediction and the experimental-frequency prediction for heads in 300 further tosses. (5)
  5. For one fair die roll, event E is “even”. List favourable outcomes, calculate P(E), find P(not E), and explain why the two probabilities sum to 1. (5)

Part C — Enrichment

11. Frequency-based prediction. In a collection of 1000 observed word tokens, “the” appears 52 times, “and” 30 times and “to” 20 times. Calculate each relative frequency. Explain how observed frequency can inform a prediction system, why a larger relevant dataset may produce a more stable estimate, why frequency alone does not guarantee the next word, and compare this with predicting sixes in repeated die rolls.

Answers

1

A sample space is the set of all possible outcomes; coin toss S={H,T}.

2

{1,2,3,4,5,6}.

3

1/2.

4

An outcome belonging to the event of interest; for “roll a 6”, the favourable outcome is 6.

5

1/3, provided the three outcomes are genuinely equally likely.

6

S={R,B,G,Y}; P(G)=1/4; expected Green=80×1/4=20. Actual frequency can differ because of random variation.

7

RF=8/60≈0.133; theoretical=1/6≈0.167. A finite random experiment can vary from the theoretical model.

8

S={A,B,C,D,E}; P(C)=1/5; expected C=20 in 100 draws. This assumes the prizes/tickets are selected with equal probability and the setup is restored for repeated trials.

9

RF=0.47. Theoretical P(H)=0.5. Theoretical prediction for 300=150; using observed 0.47 gives 141. Neither is guaranteed.

10

E={2,4,6}; P(E)=3/6=1/2; P(not E)=1/2. E and its complement cover the full sample space without overlap.

11

RF(the)=0.052, RF(and)=0.030, RF(to)=0.020. Frequencies can estimate likelihood from past observations; more relevant data can stabilise estimates. They do not make the next outcome certain, and realistic language prediction also uses context. This parallels die experiments where observed proportions can approach a theoretical probability over many trials.