← Unit 4 Experimental and Theoretical Probability

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Experimental and Theoretical Probability

⏱ 25 min📚 Year 9📈
Think First

You toss a coin 10 times and get 7 heads. Is the coin biased? What if you toss it 1000 times and get 700 heads?

💡 Revision: Ensure you understand basic probability and can convert between fractions and decimals.

Learning Intentions

Know

  • Theoretical probability
  • Experimental probability
  • Law of large numbers
  • Relative frequency

Understand

  • Why experimental probability approaches theoretical with more trials
  • When experimental probability is useful

Can Do

  • Calculate theoretical probability
  • Calculate experimental probability from data
  • Predict long-run behaviour
TheoreticalExperimentalRelative frequencyLaw of large numbersTrial
Learn Phase
1

Theoretical Probability

Based on reasoning

Theoretical probability is calculated using mathematical reasoning about equally likely outcomes.

Theoretical Probability

$P(E) = dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$

Example: P(rolling a 3 on a fair die) = 1/6 ≈ 0.167

This assumes perfect fairness, idealised models.

2

Experimental Probability

Based on data

Experimental probability (or relative frequency) is calculated from actual observations.

Experimental Probability

$P(E) approx dfrac{\text{number of times E occurs}}{\text{total number of trials}}$

Example: Rolling a die 60 times, 3 appears 8 times. Experimental P(3) = 8/60 ≈ 0.133

This may differ from theoretical probability, especially with few trials.

3

Law of Large Numbers

More trials → better estimate

The law of large numbers states that as the number of trials increases, experimental probability approaches theoretical probability.

Example: Coin tosses

  • 10 tosses: might get 7 heads (70%)
  • 100 tosses: might get 54 heads (54%)
  • 1000 tosses: likely close to 500 heads (50%)

More trials reduce the effect of random variation.

✓

Check Understanding

Try it yourself

A die is rolled 120 times. Theoretical P(6) = 1/6. How many 6s would you expect? If you actually get 15, find the experimental probability.

Worked Example

Experimental vs Theoretical

1

A coin is tossed 200 times. How many heads expected? If 95 heads occur, find experimental P(head).

Expected = $200 \times 0.5 = 100$

Experimental P(head) = 95/200 = 0.475

2

A spinner has 4 equal sections. In 80 spins, red occurs 25 times. Compare theoretical and experimental P(red).

Theoretical P(red) = 1/4 = 0.25

Experimental P(red) = 25/80 = 0.3125

Difference due to random variation with limited trials.

3

A bag has unknown marbles. After 100 draws (with replacement), 40 are red. Estimate P(red).

Experimental P(red) = 40/100 = 0.4

If the bag is large, we might estimate about 40% red marbles.

Common Misconceptions

✕

Experimental probability with few trials is very reliable. No, with few trials, random variation can produce misleading results. Need many trials for accuracy.

✕

If experimental differs from theoretical, the experiment must be unfair. Not necessarily, random variation naturally causes differences, especially with few trials.

✕

The law of large numbers guarantees exact results. No, it says experimental probability approaches theoretical; it never guarantees exact equality.

Your Turn

Practice, Experimental Probability

Work through each question in your book or digitally. Answers are in the Questions phase.

1Roll a die 60 times. Record results and compare experimental P(each number) to theoretical 1/6.
2A coin is tossed 50 times with 30 heads. Is it biased? Explain.
3Explain why casinos always win in the long run using the law of large numbers.
Real-World Anchor

Insurance and Gambling

Insurance premiums are based on experimental probability (historical data about accidents, illnesses, deaths). Gambling venues rely on the law of large numbers, while individual gamblers might win, over millions of bets the house edge guarantees profit. Australian casinos and lotteries are designed using these mathematical principles.

📓 Copy Into Your Books

▼

Theoretical

  • Based on reasoning
  • Assumes fairness
  • P = favourable/total

Experimental

  • Based on data
  • Relative frequency
  • P ≈ occurrences/trials

Law of large numbers

  • More trials → closer to theoretical
  • Reduces random variation
  • Never guarantees exact equality
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Theoretical probability is based on:
Experimental probability is also called:
Toss coin 10 times, get 7 heads. Best conclusion:
Law of large numbers says:
Die rolled 60 times. Expected sixes:
Spinner (4 sections) spun 100 times. Red appears 20 times. Experimental P(red):
Theoretical P(6 on die) =
With more trials, experimental probability:
1A die is rolled 90 times. How many of each number do you expect? If 4 appears 20 times, find the experimental probability.
2Explain the difference between theoretical and experimental probability.
3Why might a casino be unconcerned about a single player winning big?

Comprehensive Answers

190 rolls. Expected each number? Experimental P(4) if 20 fours.
Expected 15 of each. Experimental P(4)=20/90≈0.222.
2Difference between theoretical and experimental.
Theoretical: calculated from equally likely outcomes. Experimental: calculated from actual observations/data.
3Casino and single big winner.
Law of large numbers. One win is offset by millions of bets. Long-term profit is guaranteed.
MC 1Theoretical based on.
Mathematical reasoning. Answer: B
MC 2Experimental also called.
Relative frequency. Answer: B
MC 310 tosses, 7 heads.
Need more trials. Answer: B
MC 4Law of large numbers.
Experimental approaches theoretical. Answer: B
MC 5Expected sixes in 60 rolls.
60/6=10. Answer: B
MC 6Experimental P(red).
20/100=0.2. Answer: A
MC 7Theoretical P(6).
1/6. Answer: B
MC 8More trials effect.
Approaches theoretical. Answer: C
SA 190 rolls, 20 fours.
Expected 15 each. Experimental P(4)≈0.222.
SA 2Theoretical vs experimental.
Reasoning vs observation.
SA 3Casino unconcerned.
Law of large numbers ensures long-term profit.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.
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