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📖 Lesson 18 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Theoretical vs Experimental Probability

Compare theoretical probabilities with experimental results and understand the law of large numbers.

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Core learning
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Worksheet

Worksheet

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Warm-up
Think First
+5 XP each

Q1 · A fair die should land on 6 one-sixth of the time. If you roll it 10 times and never get a 6, is the die unfair or is that just chance?

Q2 · Why do casinos always win in the long run, even though any single bet might go either way?

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From the lesson
Intentions

Learning Intentions

Know

  • Theoretical probability is calculated from assumptions. Experimental probability is calculated from actual trials.

Understand

  • Why experimental probability approaches theoretical probability as the number of trials increases.

Can Do

  • Design and conduct simple probability experiments, record results, and compare with theoretical predictions.
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From the lesson
Key Terms

Key Terms

Theoretical probability — Probability calculated based on mathematical reasoning and assumptions.
Experimental probability — Probability calculated from the results of actual trials or experiments.
Relative frequency — The proportion of times an event occurs: number of successes / total trials.
Law of large numbers — As the number of trials increases, experimental probability approaches theoretical probability.
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Misconceptions

Misconceptions to Fix

✗

Wrong: Independent events cannot both occur.

✓

Right: Independent events can both occur. Independence means the outcome of one does not affect the probability of the other.

✗

Wrong: Drawing cards without replacement creates independent events.

✓

Right: Drawing without replacement creates dependent events because the probability changes after each draw.

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From the lesson
Content

Theoretical vs Experimental Probability

Work through the content, activities and worked examples below. Test your understanding with the questions in the Questions phase.

Remember Independent events: P(A and B) = P(A) × P(B). With replacement = independent. Without replacement = dependent.
Exam Tip A quick test for independence: does P(A|B) = P(A)? If yes, the events are independent.
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From the lesson
Activity
✏ Activity 1 — Test for Independence

Determine whether each pair of events is independent or dependent:

  1. Flip a coin and roll a die.
  2. Draw two cards without replacement.
  3. The weather on Monday and the weather on Tuesday.
  4. Select a student from Year 10 and select a student from Year 11.
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From the lesson
Worked Example

Worked Example

Step-by-step
A bag contains 4 red and 6 blue marbles. Two marbles are drawn with replacement. Show that the events are independent and find P(both red).
  1. 1
    Step 1: P(first red) = 4/10 = 0.4. After replacement, the bag is unchanged.
  2. 2
    Step 2: P(second red) = 4/10 = 0.4. The probability does not change.
  3. 3
    Step 3: Since P(second red | first red) = P(second red), the events are independent.
  4. 4
    Step 4: P(both red) = P(first red) × P(second red) = 0.4 × 0.4 = 0.16.
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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First response. What new understanding do you have now?

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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Multiple Choice

Multiple Choice

Select the best answer for each question.

1 mark A coin is tossed 10 times and lands on heads 7 times. The experimental P(heads) is:

1 mark The theoretical probability of rolling a 6 on a fair die is:

1 mark As the number of trials increases, experimental probability:

1 mark Experimental probability is calculated as:

1 mark A die is rolled 60 times and a 6 appears 8 times. This is:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks A coin is flipped and a die is rolled.
(a) Show that the events are independent.
(b) Find P(head and number greater than 4).
(c) Find P(tail or number less than 3).

2. 4 marks A bag contains 5 red and 5 blue marbles. Two marbles are drawn without replacement.
(a) Find P(first red).
(b) Find P(second red | first red).
(c) Explain why these events are dependent.

3. 3 marks Give a real-world example of two independent events and two dependent events. Explain your reasoning.

Marking guidance: 1 mark each for MCQs. See mark allocations for each short answer question.

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