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Probability Distribution

A probability distribution is a function that assigns a probability to each possible outcome of a random variable. It tells you how likely each value is, and the probabilities of all outcomes must add up to 1.

A probability distribution is a mathematical function that maps every possible value of a random variable to a probability. For a discrete random variable XX, the distribution is defined by a probability mass function P(X=xi)P(X = x_i) for each value xix_i, where each probability satisfies 0≤P(X=xi)≤10 \le P(X = x_i) \le 1 and the sum of all probabilities equals 1. For a continuous random variable, the distribution is described by a probability density function f(x)f(x) whose integral over the entire range equals 1.

Key Formula

∑i=1nP(X=xi)=1\sum_{i=1}^{n} P(X = x_i) = 1
Where:
  • XX = the random variable
  • xix_i = the i-th possible value of X
  • P(X=xi)P(X = x_i) = the probability that X takes the value x_i
  • nn = the total number of possible outcomes

Worked Example

Problem: A spinner has four sections labeled 1, 2, 3, and 4. The probability distribution for the outcome X is given as: P(X = 1) = 0.10, P(X = 2) = 0.25, P(X = 3) = 0.40, and P(X = 4) = 0.25. Verify this is a valid probability distribution and find the expected value of X.
Step 1: Check that every probability is between 0 and 1.
0.10,  0.25,  0.40,  0.25  ✓0.10, \; 0.25, \; 0.40, \; 0.25 \;\checkmark
Step 2: Check that all probabilities sum to 1.
0.10+0.25+0.40+0.25=1.00  ✓0.10 + 0.25 + 0.40 + 0.25 = 1.00 \;\checkmark
Step 3: Compute the expected value using the formula E(X) = Σ x_i · P(X = x_i).
E(X)=1(0.10)+2(0.25)+3(0.40)+4(0.25)E(X) = 1(0.10) + 2(0.25) + 3(0.40) + 4(0.25)
Step 4: Evaluate the sum.
E(X)=0.10+0.50+1.20+1.00=2.80E(X) = 0.10 + 0.50 + 1.20 + 1.00 = 2.80
Answer: The distribution is valid because all probabilities are between 0 and 1 and they sum to 1. The expected value of X is 2.80.

Visualization

Why It Matters

Probability distributions are the foundation of statistical inference. When you conduct a hypothesis test or build a confidence interval in AP Statistics, you rely on known distributions — like the normal or binomial distribution — to determine how likely your observed data would be under certain assumptions. They also appear in fields ranging from quality control in manufacturing to modeling risk in finance.

Common Mistakes

Mistake: Assuming every outcome must have the same probability.
Correction: Probability distributions can be uniform, but they don't have to be. Each outcome can have a different probability — the only requirement is that all probabilities are between 0 and 1 and they sum to 1.
Mistake: Confusing a probability density value with an actual probability for continuous distributions.
Correction: For continuous random variables, f(x) at a single point is not a probability. You need to integrate f(x) over an interval to obtain the probability that X falls within that range.

Related Terms