I like math, particularly algebra and probability.  This article is the first of what may become a semi-regular segment that takes a deeper look into topics of a mathematical nature.

Russian Roulette

One of the most dangerous games in the world is Russian Roulette.  The player inserts one bullet into a revolver with six chambers.  The player then spins the cylinder and pulls the trigger.

What is the chance that you can spin six times, pull the trigger each time, and hit an empty chamber every time?

The basic math of the situation is that the odds of hitting an empty chamber on any particular spin is 5/6 (5 empty chambers, 1 chamber with a bullet), or .833.

Just as the odds of having a coin come up heads X consecutive times is .5^X, this situation is .833^X.  In our case, this is .833^6, or .335.  You have a 1 in 3 chance of walking away from this game alive.

But I still wouldn’t recommend trying.

Fuel Consumption

Let’s look at these two scenarios:

A: Upgrade a car that gets 10 mpg for a car that gets 20 mpg
B: Upgrade a car that gets 20 mpg for a car that gets 30 mpg

At first glance, it appears that either scenario will result in the same amount of fuel savings, right?  After all, you’re saving 10 mpg in either case.

This isn’t the case, though.  Let’s assume 10,000 miles are driven in a year,  Scenario A results in fuel consumption dropping from 1000 gallons to 500 gallons a savings of 500 gallons per year.  Scenario B results in consumption dropping from 500 gallons per year to 333 gallons a savings of a mere 167 gallons.  Huh?  What’s the trick?

The problem is that we’re trying to use the wrong tool.  We want to determine the change in fuel consumption but the mpg is not the rate of fuel consumption.  It is the mathematical reciprocal of the rate of consumption.

Let’s take a fresh look at the two scenarios, using the actual fuel consumption rates.  We’re using the exact same cars, but simply stating the facts in a different manner.

Scenario A: Upgrade a car that consumes 0.1 gallons/mile for one that consumes 0.05 gallons/mile

Scenario B: Upgrade a car that consumes 0.05 gallons/mile for one that consumes .033 gallons/mile.

The difference becomes clear scenario A reduces fuel consumption by .05 gallons per mile and scenario B reduces fuel consumption by .0167 gallons/mile.

It almost makes you wonder why the government didn’t use fuel consumption rate in the Cash For Clunkers guidelines instead of mileage.

Why do we, as a whole, use the wrong tool to gauge fuel consumption?  Probably because we prefer to use whole numbers rather than fractions.

Pizza Pi

We’ll finish up with an easy problem.

The last time my friends came over, I ordered an 8 inch pizza.  My friends could only eat half as much pizza as they wanted before the pizza was gone.  This time, I was more prepared and ordered a 16 inch pizza but there is lots of pizza left over.  What did I do wrong?

A lot of folks in the crowd are going to immediately know the answer to this one.  The area of a circle is Pi times the square of the radius (the radius being half the diameter or 4 inches for the 8 inch pizza and 8 inches for the 16 inch).  This means that the 16 inch pizza is four times as large as the 8 inch pizza, not merely twice as large.  The 16 inch pizza has ~200 square inches [Pi X (8^2)] whereas the 8 inch pizza has ~ 50 square inches [Pi X (4^2)].