Chance
How likely is it, really? Out of how many?
Try it
balls dropped
Drop one ball and see where it lands. Then drop a hundred.
No single ball is predictable. The pile is.
The yellow line: where the balls should pile up, on average.
What just happened
Each ball met 12 pegs. At every peg it went left or right, like a coin toss. Most paths mix lefts and rights, so most blue balls end up near the middle. To reach an end bin, a ball has to go the same way 12 times running: about 1 ball in 4,096.
That is why the pile grows into the shape of the yellow line every time, even though nobody can call a single ball. Even the middle bin, the likeliest one, catches only about 23 balls in 100. Drop a few hundred more and the pile hugs the line more closely.
You cannot call one ball, but you can count on a thousand. Most chances in life work the same way. So when someone tells you how likely something is, ask the question this room is named for: out of how many?
Why the middle bin catches the most
Count the paths. A ball reaches an end bin only one way: left at all 12 pegs, or right at all 12. To reach the middle bin it needs six lefts and six rights, in any order, and there are 924 different orders.
There are 4,096 possible paths in all. 924 of them end in the middle: about 23 in 100, the share the middle bin catches. No ball is steered there. The middle simply has the most ways in.
See it as 100 people
A headline says something doubles your risk. That sounds big. Set how many people in 100 it happened to before, and how many after.
The headline: doubles your risk
In people: from 1 in 100 to 2 in 100: one more person in a hundred.
The blue people had it before. The yellow ones are the change. A relative risk (“doubles”, “50% higher”) tells you how far the number moved. An absolute risk (“2 in 100”) tells you how big it is. You need both, and you need to know who was counted, and over how long.
Cancer Research UK did this sum for processed meat in 2015. Among 1,000 people who eat the least, about 56 get bowel cancer at some point in life. Among 1,000 who eat the most, about 66. “Around 17 per cent higher” is 10 more people in a thousand. (Cancer Research UK, 2015)
Watch: 200 countries, 200 years, 4 minutes
The same habit at the size of the whole world: count everyone, over a long time, before you decide what is true.
Start with how common it is
A medical test comes back positive. Before you ask how good the test is, ask how common the condition is. Picture 1,000 people. The numbers here are examples: set them to match a real test if you know one.
Everyone who tests positive, side by side:
Of the 59 people who test positive, 9 actually have it: about 1 in 7.
The test is good, yet most positives are false alarms. A small error rate, applied to the many people who do not have it, can outnumber the true cases among the few who do. Slide “How common” up and watch the answer change while the test stays the same. So ask two things: how common is this for people like me, and what would a second test add?
Watch: the same sum, drawn as a picture
Counting through 1,000 people is Bayes’ theorem without the algebra. Here is the same idea drawn as a picture.
What it is worth on average
The store offers an extended warranty. On average it is worth the chance you will need the repair times what the repair would cost. These numbers are examples; set your own.
On average it pays back $20 for the $90 it costs.
Out of 100 buyers, the store takes in $9,000 and pays out about $2,000 in repairs. It would only break even for you if 45 in 100 buyers needed the repair.
The average leaves out one thing: how much a loss would hurt. If a surprise repair bill would be a crisis you could not cover, paying more than the average to take that risk away can be wise: that is what insurance is for. If your savings could cover the repair, the average says keep the money and set it aside.
Paying for a long shot
A clinic offers a costly treatment that might help, though it helps few of the people who try it. You cannot put a price on hope, but you can ask how big the chance would need to be. These numbers are examples; set your own.
On average it gives back $2,500 of the $15,000 it costs.
Out of 100 people who try it, about 5 are helped. It would only be worth its price to you if 30 in 100 were helped.
A dollar figure for getting better can feel wrong. It is only a way to ask: what would we pay if this were sure? Then ask the doctor for their own numbers. Out of 100 people like this one, how many were helped, how much, and for how long? If no one can say, you do not know the chance, and that is worth knowing too. Last, ask what the same money would buy for sure, such as hours of help at home, and set a date to stop if nothing has changed.
Streaks happen
Here are 20 coin flips, heads and tails. The yellow band marks the longest run of the same face in a row.
Longest streak in 100 flips
Chance is streakier than it feels. In 100 fair flips, a run of 6 or more turns up about 81 times in 100, and a run of 5 or more about 97 times in 100. So a hot streak, or a perfect score from a handful of reviews, can come from luck alone. Before you read a streak as a sign, ask how many tries there were.
Where you’ll meet this
| You meet | Ask |
|---|---|
| A headline says something “doubles your risk” | Doubles from what? Ask for both numbers out of 100. From 1 in 100 to 2 in 100 is one more person in a hundred. See it as 100 people |
| A medical test comes back positive | How common is this for people like me? Then: how often is the test wrong, and what would a second test add? Count 1,000 people |
| The store offers an extended warranty | The chance of a repair times its cost, against the price. Could your savings cover the repair instead? Work it out |
| A costly treatment that helps only a few | How many out of 100 like this one were helped, and for how long? How big would that number need to be to be worth the price? Find your number |
| The forecast says 30% chance of rain | The National Weather Service defines it as the chance of at least 0.01 inch of rain at a given point (weather.gov). Picture 10 days with that forecast: if the forecast is well made, it rains on about 3 of them. |
| A hot streak at the ballpark | Streaks turn up by chance alone. A 1985 study of basketball shooting found no “hot hand”; a 2018 reanalysis found a flaw in its method and reversed that conclusion (Miller and Sanjurjo, 2018). On its own, a streak is weak evidence: ask how many tries there were. Flip for streaks |
| Three 5-star reviews, or 400 that average 4.6 | Out of how many? Three reviews can all be five stars by luck, or by friends. Four hundred reviews tell you more, even with a lower average. |
What Scripture says
I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favor to men of skill; but time and chance happen to them all.
Ecclesiastes 9:11In context. The Teacher observes that the fastest do not always win the race; time and chance happen to everyone.
The lot is cast into the lap, but its every decision is from the LORD.
Proverbs 16:33In context. Casting lots was one way Israel sought a decision; the proverb says the outcome belongs to the LORD. It is about God’s rule over what looks random, not about luck, and it does not approve of gambling.
In the morning sow your seed, and in the evening don’t withhold your hand; for you don’t know which will prosper, whether this or that, or whether they both will be equally good.
Ecclesiastes 11:6In context. Sow in the morning and keep working in the evening, because you do not know which will succeed.
Or what king, as he goes to encounter another king in war, will not sit down first and consider whether he is able with ten thousand to meet him who comes against him with twenty thousand?
Luke 14:31In context. Jesus’ second picture, after the builder: a king who sits down to judge whether his ten thousand can meet twenty thousand. Also about counting the cost of following him.
Watch
Use it on your choice
Every choice has things that might go wrong. In the walk-through, list them, then give each one honest odds: out of 100 people in your place, how many would it happen to?