MISSION · Data and probability
The Probability of a Single Event
Turning "probably" into a number between 0 and 1
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LEVEL 2 · LEARN
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The idea in one sentence
Probability counts the outcomes you want and divides by all the outcomes there are.
\[P(\text{event}) = \frac{\text{outcomes that count as the event}}{\text{all equally likely outcomes}}\]The scale
Every probability lands between 0 and 1, and the two ends mean something definite.
| Value | Meaning |
|---|---|
| 0 | Impossible — rolling a 7 on a six-sided die |
| \(\frac{1}{2}\) | As likely as not — a fair coin landing heads |
| 1 | Certain — rolling a number below 7 on that die |
You can write the same number as a fraction, a decimal, or a percent. \(\frac{1}{4}\), 0.25, and \(25\%\) are one answer in three costumes.
The word equally likely in the formula is doing real work. Counting outcomes only works if each outcome has the same chance. A bag with 3 red marbles and 1 blue has four equally likely marbles, not two equally likely colours — which is why \(P(\text{blue}) = \frac{1}{4}\) and not \(\frac{1}{2}\).
The complement is free
If something happens with probability \(p\), it fails to happen with probability \(1 - p\). The two must add to 1, because one of them definitely occurs.
That shortcut saves work constantly. “At least one head in four flips” is tedious to count directly. “No heads at all” is a single case, and the answer is what’s left over.
Expected is not promised
A probability of \(\frac{1}{2}\) does not mean ten flips give five heads. It means that over many, many flips the fraction of heads settles near a half. Short runs wander — three heads in a row is unremarkable, and the coin has no memory of them. Over thousands of trials, the measured fraction closes in on the calculated one. That convergence is the entire reason theoretical probability is worth calculating.
Common mix-ups
- The gambler’s fallacy — believing a run of losses makes a win “due.” A fair device has no memory. The next trial ignores everything before it.
- Writing a probability above 1, or as a negative. If your answer is 1.4, you counted an outcome twice or divided by the wrong total.
- Confusing the count with the probability. Six blue marbles out of ten is a probability of 0.6, not 6.
- Expecting short runs to look average. Real randomness is streakier than people expect.
Check yourself
A bag holds 5 green and 3 yellow counters. What is the probability of drawing green? Of not drawing green?
\(P(\text{green}) = \frac{5}{8} = 0.625\). Not green is \(1 - \frac{5}{8} = \frac{3}{8}\), which matches counting the 3 yellows directly.
LEVEL 3 · REFLECT
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