Maths/Distributions

Drawing at random

After thiswhat you will be able to doTurn a finite list of shares into one random draw, and multiply probabilities only after checking that the draws are independent.

Questionwhat this lesson answersA list of shares says what could happen and how much of the total each outcome holds. How do you get one actual outcome out of it, and when does the chance of several outcomes together become a multiplication?

Not coveredwhat this lesson leaves outWe build one draw, then many. We do not derive how a machine produces an even draw in the first place. We do not prove that repeated tallies must settle toward the shares. We do not say how many draws it takes to measure an unknown share, which is the lesson on telling a real difference from noise.

You have a list of shares. North holds 0.5, east 0.25, south 0.15, west 0.1. The list tells you what could happen and how much of the total each outcome holds.

Now you need one actual direction. Not a share, a direction. Nothing said so far gets you there. A share is a number attached to an outcome, and it is not an outcome.

Laying the shares along a line

Draw a line from 0 to 1. Give each outcome a stretch of that line as wide as its share, laid end to end in order.

North runs from 0 to 0.5. East runs from 0.5 to 0.75. South runs from 0.75 to 0.9. West runs from 0.9 to 1. The four stretches cover the line exactly once, with no gap and no overlap, because the shares add to one.

Now take one number from between 0 and 1, with no part of that range favoured over any other. That is an even draw, and it is the one thing this lesson takes as given rather than builds.

The drawn number lands in exactly one stretch. That stretch names the outcome.

Try it by hand. Draw 0.31 and you land in north. Draw 0.62 and you land in east. Draw 0.91 and you land in west. Draw 0.75 and you land in south, because a number landing exactly on a boundary belongs to the stretch that starts there. Somebody has to decide that case, and this is the decision.

Notice what has been arranged rather than discovered. North comes up half the time because north was given half the line. The construction was built to make that true. It is not a finding about the world.

Repeating the draw

One draw is a trial. Do it again and you may well get something else.

This is where intuition needs correcting. A share of 0.5 does not mean the outcomes alternate. Four norths in a row is not evidence that anything is broken, and it is not evidence that west is now due. The stretches do not move between draws. Each draw meets the same four as the one before.

Press the island’s button fifty times and the tally will drift toward the shares. Press it four times and it may not resemble them at all.

Independence

Two draws are independent when knowing how the first came out tells you nothing about how the second will.

Drawing again from the same unchanged list is independent. The stretches did not move.

Now take a bag of ten tickets, four of them marked, and draw two without putting the first back. These are not independent. If the first came out marked, only three marked tickets remain among nine, so the second draw faces different shares. Knowing the first result changed what you should expect from the second.

The failing case is the one to watch. Independence gets assumed silently, and it is false more often than people notice.

Why independent chances multiply

Take two draws, each with two outcomes: success with chance 0.9, failure with chance 0.1.

Write out all four combinations and how often each turns up. Success then success happens 0.9 of the time on the first draw, and of those, 0.9 again on the second. So it happens

of the time. The other three combinations account for 0.09, 0.09 and 0.01, and those four add to 1, which is the check that nothing was lost.

The multiplication came out of the counting. It works because the second draw’s shares did not depend on how the first came out. That is what independence means, and if it fails, this arithmetic says nothing at all.

Extend it to more draws and each one contributes another factor. For ${n}$ independent attempts each succeeding with chance ${p}$, all of them succeed with chance

A near certainty is not a certainty

Suppose each of ten positions is filled correctly with chance 0.99. That is a very good chance. All ten together:

Ten steps have turned a 99 percent chance into a 90 percent chance. Now suppose two hundred steps each avoid a stumble with chance 0.97:

A 97 percent chance per step has become roughly one run in four hundred and forty. Nothing went wrong at any individual step. The steps simply accumulated.

Both of those numbers depend entirely on the steps being independent. If a mistake at one step makes the next more likely to go wrong, the product is not the answer, and the true chance is usually worse rather than better.

What this does not tell you

It does not tell you the shares were right. The whole construction takes them as given, and a list of shares that does not match how often things really happen will produce draws that are faithful to the list and wrong about the world.

It does not tell you how many draws you need to pin down a share you do not already know. Ten trials will not do it, and the reason is the subject of telling a real difference from noise, later in this topic.