The "Subtract One" Rule

Most of the expression of the odds, or chances, of something happening involves simple odds that can be expressed as an ordinary fraction. There are seven days in a week, one of those days is Tuesday. So if we choose at random a day in the past or future, there is a 1/7 chance that the day will be a Tuesday.

But, I find that odds are often more complex than that and involve what I call "The Subtract One Rule". I would like to give my version of complex odds.

Suppose that, in a town, there are 6 red cars, 3 white cars and, 25 blue cars. There is a random collision between two cars. What are the odds that the collision involved two cars of the same color (colour)?

There are 34 cars altogether. So we would, subtracting one from both numerator and denominator, multiply 6/34 x 5/33 for the red cars. Multiply 3/34 x 2/33 for the white cars. Multiply 25/34 x 24/33 for the blue cars, and then add the three products together. This gives us the odds of the collision being between cars of the same color (colour) as 636/1122, or nearly 57%.

It is necessary to subtract one from both numerator and denominator as we proceed because after we have the first car, there is one less car both in the ones of that particular colour (color), and also in the total number of cars.

An ideal example of this rule was given in the posting about granularity on the progress blog, involving pairs of gloves in a drawer. Suppose that there are ten gloves in a drawer, five right gloves mixed with five left gloves. If you reach in and pull out two gloves without looking, what will be the odds that you will have a matching pair?

Your first answer might be 50/50, or even odds of pulling out a matching pair. But this is not correct. When you take the first glove, whether it is right or left, that will leave four that will not be a match with it but five that will. So, the odds are actually 5/9 that you will pull out a matching pair. We must remember to "subtract one" from the total so that the odds are 5/9, instead of an even 5/10. The odds would be even only if there were an infinite number of gloves in the drawer.

We also must remember to subtract one in order to find the odds of the collision involving two cars of any given colour (color). There are only three white cars so that the chances of a random collision being between two white cars is a slim 3/34 x 2/33. This is because once the first car is white, there will be only two white cars remaining, so that we must remember to "subtract one". The odds of the collision involving two white cars is thus 6/1122, or just over one half of one percent.

If there were a random three-car collision, the odds against all three cars being white would be an extremely slim 3/34 x 2/33 x 1/32, or 6/35904 which is equivalent to 1.67 out of a thousand.

To find the odds of a random two-car collision involving one of the six red cars and one of the three white cars, we have to multiply fractions as well as subtracting one. The odds of a white car being the first car is 3/34. Then, the odds of the second car being a red car is 6/33. This gives us 18/1122, or about 1.6%. Notice that this is three times the odds of the collision involving two white cars because there are three times as many red cars as white cars available as the second car after subtracting the first white car.