Theoretical Probability

Predicting outcomes using mathematics, not guesswork

CAPS Grade 10 | Mathematics

The Probability Formula

P(E) = n(E) / n(S)

Where:

  • P(E) = Probability of event E
  • n(E) = Number of favourable outcomes
  • n(S) = Total number of possible outcomes

The Probability Scale

00.250.50.751
ImpossibleUnlikelyEvenLikelyCertain

All probabilities fall between 0 and 1

The Complement Rule

P(not E) = 1 - P(E)

The probability of an event NOT happening

Worked Example

Question:

A fair six-sided die is rolled. What is the probability of rolling an even number

Solution:

Step 1: Sample Space S = {1, 2, 3, 4, 5, 6} → n(S) = 6

Step 2: Event E = {2, 4, 6} → n(E) = 3

Step 3: P(E) = 3/6 = 1/2 = 0.5 = 50%

50%

Dice Roll Simulator

Click the die to roll it. Watch how the experimental probability of rolling an even number approaches 50%!

⚀
0
Total Rolls
0
Even Numbers
0%
Experimental Probability (Even)
0%

Theoretical probability of even number = 3/6 = 50%

Coin Flip Simulator

Click the coin to flip it. Watch how the experimental probability approaches 50% as you flip more!

H
0
Heads
0
Tails
0
Total Flips
0% Heads

Theoretical probability of heads = 1/2 = 50%

Card Probability Explorer

Click on cards to select them. The probability will update in real-time!

0/12 = 0
0%

Test Your Knowledge

Question 1: A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of drawing a blue marble

A) 1/2
B) 1/5
C) 3/10
D) 2/10

Question 2: What is the probability of rolling a number greater than 2 on a standard die

A) 1/3
B) 1/2
C) 2/3
D) 4/6

Question 3: If P(E) = 0.35, what is P(not E)

A) 0.35
B) 0.65
C) 0.75
D) 1.35

Key Takeaways

P(E) = n(E)/n(S) - The foundation of theoretical probability
0 ≤ P(E) ≤ 1 - Probability is always between 0 and 1
P(not E) = 1 - P(E) - The complement rule
All outcomes equally likely - This is the key assumption for theoretical probability

Key Terms

Theoretical Probability Sample Space Event Favourable Outcome Complement Random Experiment Equally Likely Probability Scale

Exam Focus: Theoretical Probability

Theoretical probability is based on what should happen if all outcomes are equally likely. Before calculating, write the sample space and count the favourable outcomes. This helps you avoid using a number from the question without checking what it represents.

Give your answer in the form the question asks for: fraction, decimal, or percentage. If no form is specified, a simplified fraction is usually the clearest answer because it shows the favourable outcomes over the total possible outcomes.

If the outcomes are not equally likely, do not use the basic theoretical formula blindly. Read the question carefully to see whether it gives frequencies, weights, or experimental results.

Back to Probability Next: Relative Frequency