### What’s the Truth in “You are 200 times more likely to die in a car crash than a plane crash!”

Warning: if you think you may be disturbed by this topic, please stop reading now!

This is a rare non-IT blog post ... just something that came up in an email conversation and piqued my curiosity.

Introduction

The statement -

“You are 200 times more likely to die in a car crash than a plane crash!”

- raises the following questions:

1) Where has this figure of 200 come from (how is this quantified)?
2) How is this qualified?

The statement by itself, without further quantification and qualification, is pretty meaningless.

Well Defined Statitistics

At this website - http://www.pbs.org/wgbh/nova/space/how-risky-is-flying.html - there is a table with data provided by the U.S. Department of Transportation. And it has the following interesting and well defined statistics:

Motor Vehicles:
1.3 deaths per 100 million vehicle miles

Air Carriers:
1.9 deaths per 100 million aircraft miles

Note: The data is from 1999-2003.

When you look at these statistics, Air Travel doesn’t look that much safer than Motor Vehicular travel!

Likeliness of Being Involved in a Fatal Car/Plane Crash

Going back to the statement in question, can we use the statistics above to reach this figure of 200?

Consider a car with one person in.
Consider a plane with 200 people in.
Consider a trip of 1000 miles.

For the car, the number of these 1000 mile journeys before its statistically likely there’ll be one fatal accident is:

(100’000’000/1.3)/1000 ~= 76’923

Note: That is nearly 77 million miles or over 3000 times around the world!

For the plane, taking into account that there are 200 people on board, the number of these 1000 mile journeys before its statistically likely there’ll be one fatal accident (in which everyone on-board dies) is:

(100’000’000/1.9)*200/1000 ~= 10’526’316

Note: That’s over 10 billion miles or over 400’000 times around the world!

And 10’526’316/76’923 ~= 136

So, in this case:

“You are 136 times more likely to die in a car crash (if you’re the only one in the car) than a plane crash (if you’re one of 200 people on board) on a journey of the same distance!”

The Formula

Notice that really the 1’000 miles above makes no difference to the calculation, so we can create a formula for:

“You are X times more likely to die in a car crash (if you’re one of Y people in the car) than a plane crash (if there’s Z people on board the plane) on a journey of the same distance!”

(100’000’000/1.9)*Z/(100’000’000/1.3)*Y = X

Or:

X = 13Z/19Y

So, Where Does the 200 Come From?

In this case X = 200 and we can calculate a value of Z that will solve the equation for different values of Y (people in the car) using the formula:

Z = X*19Y/13
Z = 200*19Y/13

For Y = 1: Z ~= 292
For Y = 2: Z ~= 585

In other words:

“You are 200 times more likely to die in a car crash (if you’re the only person in the car) than a plane crash (if there’s 292 people on board the plane) on a journey of the same distance!”

“You are 200 times more likely to die in a car crash (if you’re one of 2 people in the car) than a plane crash (if there’s 585 people on board the plane) on a journey of the same distance!”

Is There Any Truth in all This?

No, it’s just “lies, damn lies, and statistics!”

... Continuation ...

What about if we have 3 cars, each carrying 4 people?

Again, let us use a thought experiment:

Consider we have a car with 4 people in.
Consider a trip of 1000 miles.

The number of these 1000 mile journeys before it's statistically likely there’ll be one fatal accident is:

(100’000’000/1.3)*4/1000 ~= 307’692

The chance of having a fatal crash on that 1000 mile journey (in which everyone on-board dies) is therefore:

1/307’692

If we have 3 cars, the chance all 3 are involved in a fatal crash on the same journey, is 1/8 of the above.

Why 1/8?

There are only two possibilities per car - say 0 or 1; 3 cars which gives 8 possibilities (2*2*2 = 8); and the possibility of getting three 1’s is 1/8 - which is illustrated below:

000
001
010
011
100
101
110
111

So, the chance of 3 cars of 4 people being involved in a fatal crash on the same 1000 mile journey is:

1/307’692 * 1/8 = 1/2’461’536

How does the Plane Fare?

Consider we have a plane with 200 people in.
Consider a trip of 1000 miles.

The number of these 1000 mile journeys before its statistically likely there’ll be one fatal accident is:

(100’000’000/1.9)*200/1000 ~= 10’526’315

The chance of having a fatal crash on that 1000 mile journey (in which everyone on-board dies) is therefore:

1/10’526’315

How much safer is the journey in the plane?

10’526’315/2’461’536 ~= 4