What Is a Probability Distribution?

A probability distribution describes how the probabilities of different outcomes in an experiment are spread out. It tells us:

• What outcomes are possible.

   

• How likely each outcome is.

   

Example: When rolling a fair die, each number (1 to 6) has a probability of 1/6. Listing out all the possible outcomes and their probabilities gives us the probability distribution of the die roll.

Key Properties of a Probability Distribution

Every probability distribution must satisfy two main rules:

1. Probabilities cannot be negative: P(x) ≥ 0 for every possible outcome.

    

2. The total probability must equal 1: Sum of all P(x) = 1.

    

Types of Probability Distributions

(a) Discrete Probability Distributions

• Outcomes are countable (e.g., number of heads in 3 coin tosses).
• Examples: Binomial distribution, Poisson distribution.

   

(b) Continuous Probability Distributions

• Outcomes can take any value within a range (uncountable).
• Probabilities are found using areas under a curve.
• Examples: Normal distribution, Exponential distribution.

   

Example of a Discrete Distribution

Let X = number of heads when tossing 2 fair coins.

• Possible values: 0, 1, 2.

   

• Distribution:
  • P(X = 0) = 1/4
  • P(X = 1) = 2/4
  • P(X = 2) = 1/4
    

     

Example of a Continuous Distribution

Normal distribution:

• Mean (µ) = centre of the curve.
• Standard deviation (σ) = spread of the curve.
• Probabilities are found by calculating the area under the curve.

   

Example: If heights of students follow a normal distribution with mean = 170 cm and standard deviation = 6 cm, we can calculate the probability of a student being taller than 180 cm using Z-scores.

Mean and Variance of a Probability Distribution

• Mean (Expected Value, E[X]): The “average” outcome.
  • Discrete: E[X] = Σ (x * P(x))
  • Continuous: E[X] = ∫ x f(x) dx
    
    
• Variance (Var(X)): How spread out the outcomes are around the mean.
  • Formula: Var(X) = E[X²] – (E[X])²

     

Why Is Probability Distribution Important?

• Models real-world randomness (e.g., exam scores, waiting times, stock prices).
  
  
• Foundation for statistics, hypothesis testing, and A Level probability.
  
  
• Commonly tested in exams with Binomial, Normal, and Poisson distributions.