How can I deduce the probability of auto death from a death rate per billions of miles?

Question

I am working through Introductory Statistics for Business and Economics by the brothers Wonnacott, and am stuck on this problem in chapter 3 (problem 3-40):

For various forms of transportation, the 1975-78 U.S. death rates were approximately as follows (deaths per billion passenger miles):

Car: 16

Train: 0.84

Scheduled airline: 0.35

b) Suppose you travel about 20,000 miles per year. Over a remaining lifetime of 60 years, what is your approximate chance of being killed in an accident if you traveled always by car? Always by plane? By car or plane, 50-50?

c) What assumptions did you make in b?

It is easy enough to calculate the number of miles you travel in 60 years: 1.2 billion. It's also easy enough to calculate the auto death rate in that total: 16 * 1.2 = 19.2. But, I'm struggling with the transition from a rate per billion passenger miles to an approximate probability. It feels more intuitive to count deaths per passengers rather than deaths per passenger miles.

And yet, earlier in the chapter, the book defines probability as:

$$ Probability \equiv lim \left(\frac{f}{n}\right) $$

If we take $n$ to be "passenger miles" and $f$ to be "deaths," then it seems like you could assert the probability of dying in an auto accident in 1.2 billion miles is just:

$$ \left(\frac{16}{1e9}\right) * 1.2 = 1.92e-8 $$

Is that all there is to it? It makes some intuitive sense. The probability is slightly larger than the death rate because you are traveling farther. We're obviously assuming the death rate is constant, the mileage per year is constant, all the miles have the same probability of death, etc. But, it feels like it could be correct.

Answer 1

I think you are missing some zeros here. First of all, 20,000 miles travelled in 60 years is 1.2 million, not 1.2 billion. Second, I think the assumption which is hinted at in part b) is about how many passengers drove the 1 billion passenger miles. Were there 16 passengers driving 62,500,000 miles and all of them died? Probably not.

The simplest assumption would probably be that each passenger drove exactly 1.2 million miles, since then the probability of death is $$ \frac{16}{1.9 * 10^9 \, / \, 1.2 * 10^6} = 0.0192 = 1.92 \% $$ This is almost what you got except for the missing 10^6 and it implies that 16 out of approximately 833.33 passengers died.

Another meaningful assumption would be that 50,000 passengers drove exactly 20,000 miles. Then you get the probability of dying in one year and have to aggregate for the full life time. But this is more complex and the result is hardly different (I got approximately 0.0190.)