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Experimental and Theoretical Probability
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?
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
Theoretical Probability
Theoretical probability is calculated using mathematical reasoning about equally likely outcomes.
$P(E) = dfrac{ ext{number of favourable outcomes}}{ ext{total number of possible outcomes}}$
Example: P(rolling a 3 on a fair die) = 1/6 ≈ 0.167
This assumes perfect fairness — idealised models.
Experimental Probability
Experimental probability (or relative frequency) is calculated from actual observations.
$P(E) approx dfrac{ ext{number of times E occurs}}{ ext{total number of trials}}$
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.
Law of Large Numbers
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
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.
Experimental vs Theoretical
A coin is tossed 200 times. How many heads expected? If 95 heads occur, find experimental P(head).
Expected = $200 imes 0.5 = 100$
Experimental P(head) = 95/200 = 0.475
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.
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.
Practice — Experimental Probability
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