Probability • Topic 1 of 3

Experimental and Theoretical Probability

What is Probability?

Probability is the measure of how likely an event is to occur. It is a number between 0 and 1 (or 0% and 100%). A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.

Two Types of Probability:

TypeDefinitionFormulaExample
Experimental ProbabilityBased on actual experiments or observations\(\frac{\text{Number of times event occurs}}{\text{Total number of trials}}\)Flipping a coin 50 times and counting heads
Theoretical ProbabilityBased on mathematical reasoning without experiments\(\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)Calculating probability of rolling a 3 on a fair die

Key Vocabulary:

  • Experiment: A repeatable process that gives results (e.g., rolling a die, flipping a coin)
  • Outcome: A possible result of an experiment (e.g., heads, tails, rolling a 4)
  • Event: A set of outcomes we are interested in (e.g., rolling an even number)
  • Sample Space: The set of all possible outcomes (e.g., {1, 2, 3, 4, 5, 6} for a die)
  • Trial: Each repetition of an experiment

Comparing Experimental and Theoretical Probability:

AspectExperimentalTheoretical
Based onActual data from experimentsMathematical reasoning
AccuracyImproves with more trialsExact for ideal situations
RequiresPerforming the experimentKnowing all possible outcomes
ExampleFlipped coin 100 times, got 48 heads (0.48)Fair coin always has 0.5 probability of heads

Law of Large Numbers: As the number of trials increases, experimental probability gets closer to theoretical probability.

Real-life Applications:

  • Weather forecasting (probability of rain)
  • Insurance companies (probability of accidents)
  • Sports analytics (probability of winning)
  • Board games (probability of rolling certain numbers)
Experimental vs Theoretical Probability0Impossible0.25Unlikely0.5Even0.75Likely1CertainTHEORETICALEXPERIMENTALBased on equal likelihoodP(event) = favourable/totalCoin: P(H) = 1/2 = 0.5Calculated before doingNo need to conduct trialBased on actual resultsP(event) = times happened/total trials100 flips: 48 headsP(H) = 48/100 = 0.48More trials → closer to theoretical
Solved Examples
1
Worked Example

A coin is flipped 200 times. Heads appears 108 times. Find the experimental probability of getting heads. Also, state the theoretical probability.

Solution
  • Step 1: Experimental probability = \(\frac{\text{Number of heads}}{\text{Total flips}}\)
  • Step 2: = \(\frac{108}{200} = \frac{54}{100} = 0.54\)
  • Step 3: Theoretical probability = \(\frac{1}{2} = 0.5\) (since fair coin has 2 equally likely outcomes)

Answer: Experimental probability = 0.54, Theoretical probability = 0.5

2
Worked Example

A bag contains 4 red, 3 blue, and 5 green marbles. Find the theoretical probability of drawing:

(i) a red marble

(ii) a blue marble

(iii) a yellow marble

Solution
  • Step 1: Total marbles = \(4 + 3 + 5 = 12\)
  • Step 2: P(red) = \(\frac{\text{Number of red}}{\text{Total}} = \frac{4}{12} = \frac{1}{3}\)
  • Step 3: P(blue) = \(\frac{3}{12} = \frac{1}{4}\)
  • Step 4: P(yellow) = \(\frac{0}{12} = 0\) (impossible event)

Answer: P(red) = $\frac{1}{3}$, P(blue) = $\frac{1}{4}$, P(yellow) = 0

3
Worked Example

A die is rolled 120 times. The number 5 appeared 25 times. Compare experimental and theoretical probability of rolling a 5. How does the law of large numbers apply here?

Solution
  • Step 1: Experimental P(5) = \(\frac{25}{120} = \frac{5}{24} \approx 0.2083\)
  • Step 2: Theoretical P(5) = \(\frac{1}{6} \approx 0.1667\)
  • Step 3: Difference = \(0.2083 - 0.1667 = 0.0416\)
  • Step 4: This is a small difference (about 4%)
  • Step 5: Law of large numbers says: With more trials (120 is moderate), experimental probability should get closer to theoretical
  • Step 6: If we rolled 1000 times, we would expect P(5) to be even closer to \(\frac{1}{6}\)

Answer: Experimental P(5) ≈ 0.2083, Theoretical P(5) ≈ 0.1667. The difference is small and would decrease with more trials.

Key Points

  • Probability measures likelihood from 0 (impossible) to 1 (certain)
  • Experimental probability = (event occurrences) ÷ (total trials)
  • Theoretical probability = (favorable outcomes) ÷ (total possible outcomes)
  • Law of Large Numbers: More trials = experimental probability approaches theoretical probability
  • Sample space is the set of all possible outcomes
  • Probability can be expressed as fraction, decimal, or percentage
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Theoretical probability $=$
Explanation: Favourable over total.
Q2.Experimental probability $=$
Explanation: Based on actual trials.
Q3.A probability always lies between:
Explanation: $0\le P\le1$.
Q4.$P(\text{certain event})=$
Explanation: $1$.
Practice more MCQs → 📝 Assignment · 35 marks