Statistics From Zero · Lesson 4 of 22
Probability Without Tears
In one sentence: Probability is a number between 0 and 1 for how often something happens in the long run, and the coin, the dice and the roulette wheel have no memory, so five reds in a row never make black 'due'.
The idea
A probability is a number between 0 and 1 that says how likely something is. 0 means it never happens. 1 means it always happens. Everything else is in between, and people usually write it as a percentage: 0.5 is 50%, "as likely as not".
The most useful way to think about it: a probability is how often something would happen if you could repeat the situation many, many times. A 1 in 6 chance doesn't mean it happens exactly once every six tries. It means that over thousands of tries, it happens about one time in six.
When every outcome is equally likely, working it out is just counting:
probability = the outcomes you're interested in ÷ all the possible outcomes
A die has six faces and one of them is a six, so the chance of rolling a six is 1 in 6, about 17%.
Three rules that do most of the work
- OR: add. If two things can't both happen at once, the chance of one or the other is the two chances added up. Rolling a 5 or a 6: 1/6 + 1/6 = 2/6, which is 1 in 3.
- AND: multiply. If two things don't affect each other, the chance of both happening is the two chances multiplied. Two dice, both sixes: 1/6 × 1/6 = 1/36, under 3%. That's why "and" always makes things less likely.
- NOT: take it away from 1. The chance something doesn't happen is 1 minus the chance it does. Not rolling a six: 1 minus 1/6 = 5/6.
The NOT rule hides the best trick in the lesson. Questions about "at least one" are hard to count directly, so flip them around: the chance of at least one six in four rolls is 1 minus the chance of no sixes in four rolls. No six four times in a row is 5/6 × 5/6 × 5/6 × 5/6 = 625/1296, about 48%. So at least one six is about 52%. Slightly better than a coin flip, which is the kind of thing gamblers were arguing about in the 1600s, and the reason probability got invented in the first place.
Uncle Gary at the casino
The family is in Las Vegas for cousin Kevin's birthday. Uncle Gary is standing at the roulette table, watching. The ball has landed on red five times in a row.
"Black is due," says Gary, and puts the rest of his drink budget on black.
An American roulette wheel has 38 pockets: 18 red, 18 black and 2 green. The chance of black on any spin is 18 out of 38, about 47.4%. After five reds in a row, the chance of black on the next spin is... 18 out of 38, about 47.4%. The wheel doesn't know what it did before. It's a piece of wood with a ball in it.
Black did not come. Gary spent the rest of the evening explaining to Aunt Linda that he had been "statistically robbed".
Believing that a result is "due" because it hasn't happened for a while is called the gambler's fallacy. It comes from a real idea that people get backwards.
Chance of black on the next spin
The law of large numbers (and what it doesn't say)
The real idea is the law of large numbers: over many, many spins, the share of reds will settle close to 47.4%. That's true. But it doesn't happen because the wheel "corrects" a run of reds by producing extra blacks later. It happens because, after thousands of spins, a little run of five gets lost in the crowd. The streak isn't paid back. It's just drowned out.
Spins, coin flips and dice rolls are independent events: one result doesn't change the chances of the next. Most real-life runs of luck work the same way, which is why streaks turn up much more often than people expect. Five reds in a row has about a 2.4% chance on any particular five spins, roughly 1 in 42. A busy roulette table does a lot more than 42 spins a night.
Why anyone outside class cares
Imagine a trading strategy that wins 40% of its trades. Assuming each trade is independent of the last, the chance that any particular five trades are all losers is 0.6 × 0.6 × 0.6 × 0.6 × 0.6, about 7.8%. Over a few hundred trades, a run like that is close to guaranteed. A losing streak on its own doesn't prove a strategy is broken, and it certainly doesn't mean a winner is "due". The only thing that answers the question is a lot more data, which is what our piece on bad-day streaks is about.
Try it yourself
Uncle Gary is planning a barbecue. The forecast says a 50% chance of rain on Saturday and a 50% chance of rain on Sunday. "So it's 100% going to rain at some point this weekend," says Gary, and cancels.
What's the real chance of rain on at least one of the two days? (Assume Saturday's weather doesn't affect Sunday's.)
Show the answer
Gary used the OR rule, but the OR rule only works for things that can't both happen. It can easily rain on both days, so adding is wrong. (If it were right, three days at 50% would give a 150% chance of rain.)
Use the "at least one" trick instead. The chance of no rain on Saturday is 50%, and no rain on Sunday is 50%. No rain on both days: 0.5 × 0.5 = 0.25, or 25%.
So the chance of rain on at least one day is 1 minus 0.25 = 75%. High, but there's still a 1 in 4 chance of a dry weekend that Gary just cancelled.
One honest footnote: real weather isn't fully independent. A rainy Saturday makes a rainy Sunday a bit more likely. That's the assumption we made, and naming your assumptions out loud is half of doing probability properly.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Experiment (or trial) | Anything with an uncertain result: a roll, a spin, a flip |
| Sample space | The list of everything that could happen (the six faces of a die) |
| Event | The outcome or group of outcomes you're asking about ("a six", "an even number") |
| P(A) | Just shorthand for "the probability of A" |
| Complement | "Not A". Its probability is 1 minus P(A) |
| Mutually exclusive | Can't both happen at once (a 5 and a 6 on one roll). Only then can you simply add |
| Independent events | One doesn't change the chances of the other (two separate rolls). Only then can you simply multiply |
| Addition rule / multiplication rule | The OR rule and the AND rule from this lesson |
| Law of large numbers | Over many tries, the share of outcomes settles close to the true probability |
| Gambler's fallacy | Believing a result is "due" because it hasn't happened lately |
Three things to remember
- A probability is how often something happens in the long run, from 0 (never) to 1 (always).
- OR means add (only if they can't both happen). AND means multiply (only if they're independent). "At least one" means 1 minus "none".
- Coins, dice and roulette wheels have no memory. Nothing is ever "due", and streaks are normal.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.