Probability and statistics

What is probability?

Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). For an event where every outcome is equally likely, the probability of something happening is: (number of outcomes where it happens) ÷ (total number of possible outcomes).

A basic calculation: rolling a die

A standard die has 6 faces, each equally likely. The probability of rolling a 4 is 1 ÷ 6, since only one of the six faces shows a 4. The probability of rolling an even number is 3 ÷ 6 = 1/2, since three faces (2, 4, 6) count as even out of six total.

Independent events: why you multiply

Two events are independent if one doesn't affect the other's outcome — like flipping a coin twice. Here's why you multiply their probabilities together to find the chance both happen.

Flip a coin twice. There are 4 equally likely outcomes in total: heads-heads, heads-tails, tails-heads, tails-tails — you can list every combination of the first flip (2 options) with the second flip (2 options), giving 2 × 2 = 4 total combinations. Only 1 of those 4 outcomes is heads-heads. So the probability of getting heads twice is 1/4 — which is exactly 1/2 × 1/2, the probability of heads on the first flip multiplied by the probability of heads on the second. This isn't a coincidence: whenever two independent events each have several equally likely outcomes, multiplying the individual probabilities always gives the correct combined probability, because it's really just counting all the combinations, the same way 2 × 2 counted the coin flip combinations.

What is statistics?

While probability starts from known rules and predicts what should happen, statistics works the other way: it starts from real, observed data and tries to summarize and understand it.

Three ways to summarize a set of numbers

Given a set of data — say, the ages of 5 people: 22, 25, 25, 30, 48 — there are three common ways to describe a "typical" value:

Mean (average): add all the values and divide by how many there are. (22+25+25+30+48) ÷ 5 = 150 ÷ 5 = 30.

Median: sort the values and pick the middle one. Sorted: 22, 25, 25, 30, 48 — the middle value is 25.

Mode: the value that appears most often. Here, 25 appears twice, more than any other value, so the mode is 25.

Notice the mean (30) is pulled higher than the median (25) by the one unusually large value, 48 — this is exactly why the median is often reported alongside the mean: it isn't as affected by a few extreme values.

Why probability and statistics matter

Probability lets us reason carefully about uncertainty and chance — from games and insurance to weather forecasts. Statistics lets us make sense of real-world data — from medical trial results to economic trends — and the two fields work together constantly: statistics uses probability to judge how much confidence to place in what the data seems to show.