Counting the Same Thing Twice

Last timeCounting Turns Into Chance

Two errors account for most wrong counts: counting an outcome more than once, and forgetting that two groups overlap. Both have a standard repair and both are detectable on a small case.

Almost every wrong count in practice is one of two mistakes. This lesson is

both of them, their repairs, and how to notice before anybody else does.

The first mistake: counting it twice

The previous lesson ended on one. Counting hands with two pairs, somebody picks

a first rank in thirteen ways and a second rank in twelve, getting one hundred

and fifty six pairs of ranks where the true number is seventy eight. Every pair

was counted twice, once in each order, because the procedure distinguished a

first and a second rank while the problem does not.

FIG 1Does an order belong in this count?

	the naive count	the overcount factor	the right answer	is the order real?
a captain then a vice ca	90	1	90	1
two co captains, from 10	90	2	45	0
three medals, from 10	720	1	720	1
a committee of three, fr	720	6	120	0
two ranks for the pairs,	156	2	78	0
seat 4 people in a row	24	1	24	1
seat 4 people around a r	6	4	6	0

The first two rows are the lesson in miniature. Ninety is the count either way, and whether to halve it depends entirely on whether the two positions mean different things. A captain and a vice captain are different jobs, so ninety stands. Two co captains are the same job, so every pair was counted twice and the answer is forty five. Nothing in the arithmetic distinguishes these two problems, and only reading the words does. The last row is the round table, where the naive twenty four is divided by the four rotations that produce the same seating, and the nought in the last column means the order is not part of the outcome rather than meaning nothing.

The repair is division, and the condition has now been stated three times in

this course because it is the thing that goes wrong: every outcome must have

been overcounted by the same factor. Say the factor aloud with its reason

before dividing. Every pair of ranks arises in exactly two orders, so divide by

two. Every committee of three arises in exactly six orders, so divide by six.

When the factor varies between outcomes, no division works and the problem

splits into cases.

The lesson stops here

3 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Counting the Same Thing Twice

The first mistake: counting it twice

3 more paragraphs to go

This is the reading half

Read alongside