Home › Math Lessons › Statistics › Probability Basics
StatisticsBeginner

Probability Basics

Probability is the mathematics of uncertainty. It gives you a precise way to quantify how likely something is to happen — from predicting coin flips to understanding insurance, medical tests, and financial risk.

In this lesson

  1. What probability means
  2. Calculating basic probability
  3. The addition and multiplication rules
  4. Conditional probability
  5. Real-world applications

1 What Probability Means

Probability is a number between 0 and 1 that measures how likely an event is. 0 means impossible. 1 means certain. 0.5 means equally likely to happen or not happen. Everything else falls between.

The sample space is the set of all possible outcomes. An event is any subset of the sample space. Rolling a die: the sample space is (1, 2, 3, 4, 5, 6). The event "rolling an even number" is (2, 4, 6).

Frequentist interpretation

Probability can be thought of as long-run frequency. If you flip a fair coin 10,000 times, you'll get heads very close to 5,000 times — approximately 50%. For a single flip, it's either heads or tails, but the probability describes the expected proportion over many repetitions.

2 Calculating Basic Probability

For equally likely outcomes: P(event) = Number of favorable outcomes / Total number of outcomes.

Basic Probability
A standard die is rolled. What is the probability of rolling a number greater than 4?
1Favorable outcomes: {5, 6} — 2 outcomes
2Total outcomes: {1, 2, 3, 4, 5, 6} — 6 outcomes
3P(>4) = 2/6 = 1/3 ≈ 0.333 or 33.3%
Answer: P(>4) = 1/3 ≈ 33.3%

Probabilities can be expressed as fractions, decimals, or percentages — they're all equivalent. The sum of probabilities of all possible outcomes must always equal 1 (or 100%).

3 The Addition and Multiplication Rules

Addition rule (OR): P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap to avoid double-counting.

For mutually exclusive events (can't both happen): P(A or B) = P(A) + P(B). No overlap to subtract.

Addition Rule
A card is drawn from a standard deck. What is P(red OR face card)?
1P(red) = 26/52 = 1/2
2P(face card) = 12/52 = 3/13
3P(red AND face card) = 6/52 (red face cards: J♥,Q♥,K♥,J♦,Q♦,K♦)
4P(red OR face card) = 26/52 + 12/52 − 6/52 = 32/52 = 8/13 ≈ 61.5%
Answer: P(red OR face card) = 8/13 ≈ 61.5%

Multiplication rule (AND): For independent events (one doesn't affect the other): P(A and B) = P(A) × P(B).

Multiplication Rule
What is the probability of flipping heads twice in a row?
1P(heads on flip 1) = 1/2
2P(heads on flip 2) = 1/2 (independent of flip 1)
3P(both heads) = 1/2 × 1/2 = 1/4 = 25%
Answer: P(HH) = 1/4 = 25%

4 Conditional Probability

P(A|B) reads "the probability of A given that B has occurred." Knowing B has happened changes (conditions) the probability of A.

Formula: P(A|B) = P(A and B) / P(B)

Conditional Probability
In a class, 60% play sport, 40% play music, and 25% play both. Given a student plays sport, what's the probability they also play music?
1P(sport) = 0.60, P(music|sport) = P(both) / P(sport)
2P(music|sport) = 0.25 / 0.60 = 0.417 ≈ 41.7%
Answer: 41.7% of sport players also play music
Independence vs mutual exclusivity

Independent events: one doesn't affect the other's probability (coin flips). Mutually exclusive events: they can't both happen (rolling a 3 AND a 5 on one die). These are different concepts. Mutually exclusive events are not independent — if one happens, the other's probability drops to zero.

5 Real-World Applications

Medical testing: if a disease affects 1% of people and a test is 99% accurate, the probability that a positive test means you have the disease is not 99% — it's actually around 50%, due to the large number of false positives from healthy people. This is Bayes' theorem, and misunderstanding it leads to serious medical interpretation errors.

Insurance: premiums are priced based on probability of claims. A car insurer estimates the probability you'll make a claim and the expected cost, then charges slightly more than that to cover overhead and profit. Probability underpins all of actuarial science.

Gambling: every casino game is designed so the house has a positive expected value. The probability of each outcome, combined with the payouts, guarantees the casino profits in the long run — regardless of any individual session's result.

Practice Problems

A bag has 4 red, 6 blue, and 2 green marbles. What is P(drawing a blue marble)?
P(blue) = 6/12 = 1/2 = 50%
What is P(rolling a 2 OR a 5) on a standard die?
These are mutually exclusive. P(2) + P(5) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 33.3%
A fair coin is flipped 3 times. What is P(all three are tails)?
P(T)³ = (1/2)³ = 1/8 = 12.5%
P(A) = 0.4, P(B) = 0.3, P(A and B) = 0.1. Find P(A or B).
P(A or B) = 0.4 + 0.3 − 0.1 = 0.6 = 60%