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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.

Today's hook: Theory says heads comes up half the time, so why did you just flip 7 heads out of 10?
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Core learning
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Worksheet

Worksheet

Use the worksheet to complete this lesson in your book or digitally.

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 probabilityProbability calculated based on mathematical reasoning and assumptions.
Experimental probabilityProbability calculated from the results of actual trials or experiments.
Relative frequencyThe proportion of times an event occurs: number of successes / total trials.
Law of large numbersAs the number of trials increases, experimental probability approaches theoretical probability.
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From the lesson
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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