## Tuesday, March 23, 2010

### Paul Bloom and Seatbelts

Today’s question comes from special guest Paul Bloom.  Dr. Bloom is a professor in the Psychology Department at Yale University.  He has authored several books including his most recent, How Pleasure Works, which will be published this June.  Dr. Bloom wants to know,

“How much time does the average person spend doing up his seatbelt? How does it compare to the time lost by summing up the increased chance of death and injury by not doing up one's seatbelt?”

Most of the time, it only takes about 2 seconds to buckle or unbuckle a seatbelt1.  If you drive to and from work or school each day, then you will buckle/unbuckle at least 4 times per day.  By using these numbers and assuming an 80-year lifespan, we can compute the total amount of time spent buckling/unbuckling a seatbelt in one’s lifetime,

total time = (buckles per day) · (time per buckle) · (days per year) · (years)
= (4 buckles per day) · (2 s per buckle) · (365 days per year) · (80 years)
= 2.7 days.

According to Book of Odds, the odds of dying in a transportation accident this year are about 1 in 6,279.  This being the case, the odds of not dying in a car accident this year are then 6,278 out of 6,279.  By again assuming an 80-year lifespan, we can compute the odds of never dying in a car accident,

prob. of never dying in accident = (prob. of not dying in accident this year)# of years
(6,278 / 6,279)80 = 98.7%.

This means our chances of dying in a car accident are 1.0-0.987 = 1.3%.  About half of these deaths happened to people that weren’t wearing seatbelts.  Because half the deaths occurred to people wearing seatbelts, you might be tempted to think that seatbelts don’t protect you, but that’s not the case because people wear seatbelts 83% of the time.  As an example, let’s say there are 10,000 people.  Of these people, 17% or 1700 people don’t wear a seatbelt and 83% or 8300 people do wear a seatbelt.  Of the 10,000 people, 1.3% or 130 people die in an auto accident.  Half of this 130 or 65 people were not wearing a seatbelt and the other 65 people were.  This means the probability of dying without a seatbelt is given by,

prob. of dying w/o seatbelt = (# of deaths w/o seatbelt) / (# of people w/o seatbelt)
= (65 deaths without seatbelt) / (1700 people without seatbelt)
= 3.8%.

Likewise, the probability of dying with a seatbelt is given by,

prob. of dying with seatbelt = (# of deaths with seatbelt) / (# of people with seatbelt)
= (65 deaths with seatbelt) / (8300 people with seatbelt)
= 0.8%

As you can see, the probability of dying without a seatbelt is about 4 to 5 times greater.

If we assume that people of all ages are equally likely to die in a car accident2, then we can estimate that on average a person loses about 40 years of his/her life after dying in a car accident.  From this, we can calculate the average difference in life expectancy between people that do and don’t use seatbelts,

average time lost =
(prob. of death w/o seatbelt – prob. of death with seatbelt) · (average time lost per death)
= (0.038-0.008) · (40 years lost per death)
= 1.2 years

As you can see, for a total cost of 2.7 days spent buckling and unbuckling your seatbelt, you’re expected to gain over a whole year on your life expectancy, so buckle up!

Dr. Bloom, thanks for a fun and challenging question.

[1] I say “Most of the time” because we’ve all been in a cab, rent-a-car, or some other unfamiliar vehicle where it can take upwards of a minute to figure out how to buckle a seatbelt.  I’m assuming these situations are rare enough that we can neglect them in this calculation.
[2] As I found out later, this turns out not to be the case