Probability Basics

Introduction

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage. Probability helps with decisions about weather, games, health tests and risk.

Counting equally likely outcomes

If all outcomes are equally likely:

probability = (number of favourable outcomes) / (total number of outcomes)

• A fair coin: P(heads) = 1/2 = 0.5.
• A fair die: P(rolling a 4) = 1/6.
• A fair die: P(even number) = 3/6 = 1/2.

Example with two dice: there are 6 x 6 = 36 equally likely pairs. The pairs that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), which is 6 outcomes. So P(sum 7) = 6/36 = 1/6, about 0.167.

The complement rule

The probability that something does not happen is 1 minus the probability that it does.

P(not 6 on a die) = 1 - 1/6 = 5/6.

This is useful for "at least one" questions. P(at least one head in 3 flips) = 1 - P(no heads) = 1 - (1/2)^3 = 1 - 1/8 = 7/8.

Independent events

Events are independent if one does not affect the other. For independent events, multiply the probabilities for "and":

P(two heads in a row) = 1/2 x 1/2 = 1/4.

For "or" with events that cannot both happen (mutually exclusive), add: P(rolling 1 or 2) = 1/6 + 1/6 = 1/3.

Without replacement

If you draw from a bag without putting items back, probabilities change. A bag with 3 red and 2 blue balls. P(two reds in a row) = 3/5 x 2/4 = 6/20 = 3/10. The second draw has fewer balls left.

Expected value

The expected value is the long-run average: sum of value times probability. A game pays 6 if a die shows a 6 and costs 2 to play. Expected winnings = 6 x 1/6 = 1, so you expect to lose 1 per play on average (1 minus 2). Many games of chance have a negative expected value for the player.

Common mistakes

• Gambler's fallacy: after five heads, a fair coin is not "due" tails. The next flip is still 1/2.
• Confusing "unlikely" with "impossible." Rare things happen, especially with many chances.
• Treating dependent events as independent.
• Ignoring the base rate. The final lesson in this course shows how this affects medical tests.

Real-world probabilities are often estimated from data and have uncertainty. Weather forecasts saying "30 percent chance of rain" mean that, in similar situations, it rains about 3 times in 10, according to the forecasting method.

Try it yourself

Find P(a sum of 9 with two dice), P(at least one six in two rolls) and P(two aces from a deck of 52 without replacement). Check the last with 4/52 x 3/51.

Summary

• Probability runs from 0 to 1 and is favourable outcomes divided by total equally likely outcomes.
• Use the complement rule for "at least one", multiply for independent "and", and adjust when sampling without replacement.
• Avoid the gambler's fallacy and remember that expected value describes long-run averages.