math quiz!

[quarrel]

Twenty points per question, five questions. Partial credit is possible. No time limit.

1. 90% of riders involved in motorcycle crashes have not taken a motorcycle safety training course. How much does taking a motorcycle training course lower your chance of being involved in a crash?

2. You're playing a fantasy RPG and have a choice between two swords. Sword A deals 10 damage and can be swung every 2 seconds. Sword B deals 14 damage but takes 3 seconds to swing. Which is better?

3. It's 78 degrees in Beaverton. Ninety miles away, in Salmon Falls, it's 72 degrees. Oakmont lies directly at the midpoint on a line between those two cities. What is the temperature there?

4. Suppose that last year there were 600 fatalities from car-related accidents. Also suppose that only 50% of all car occupants wore seat belts. How many fatalities will there be next year from car-related accidents if a mandatory seat belt law is passed?

5. The weather report says there's a 40% chance of rain tomorrow at noon, 45% at 3:00 pm and 50% at 6:00 pm. What is the chance that it rains tomorrow?

Comments

[quarrel]

As Shaterri points out, this isn't really a math test. I've been mentally accumulating simple questions, ideally based on real life, that show off how easy it is to make assumptions without realizing it. You all aren't quite the intended audience, and you pretty much sussed it all out. Still, it was useful to see what I missed or how I might explain things better.

And with that, on to the answers!

1. 90% of riders involved in motorcycle crashes have not taken a motorcycle safety training course. How much does taking a motorcycle training course lower your chance of being involved in a crash?

Answer: There is not enough information to answer this question.

I did not make up this 90% factoid. It's been circulating in motorcycle communities for more than twenty years. I personally heard it in college from housemates in a riding club, and I can still find it today on the web (like here and here). Look at this error those pages make:

"only 10% of those who have had formal training are involved in a motorcycle crash."

That's just plain wrong. The factoid does not say 10% of trained riders crash. It says 10% of crashed riders had training. Those aren't interchangeable. The implication is that training makes you 1/9th as likely to crash, but that's based on misinterpreting the numbers.

To actually answer the original question, you also need to know what percentage of riders in general have had formal training, then compare it against the percentage of crash victims who had training. (You could also consider just the riders who've never crashed. That works too.) If the general percentage is higher than the crash victim percentage, it's reasonable to conclude that training makes you safer. But the general percentage might be the same (which means training doesn't help) or even lower (which means training makes you less safe).

The factoid's original source seems to be the 1981 Hurt Report on U.S. motorcycle safety. The report's data were collected from 1976-1977 in and around Los Angeles. Its details are probably outdated now, but the report did discover that 8% of crash victims had training, whereas 15.7% of riders in general were trained, implying that training cuts your odds of crashing roughly in half. On the other hand, a similar international study in Europe in 2000 showed that there, trained riders accounted for a slightly larger fraction of crash victims (59.9%) than of riders in general (51.6%). So taking a professional course in motorcycle handling over there apparently didn't make you any safer than learning how to ride from friends or relatives. In fact, it might even be a bad idea.

2. You're playing a fantasy RPG and have a choice between two swords. Sword A deals 10 damage and can be swung every 2 seconds. Sword B deals 14 damage but takes 3 seconds to swing. Which is better?

Answer: There is not enough information to answer this question.

First, "Which is better?" means "Which kills things faster?" That is an assumption, but it's a safe one in context. But you also need to know how many hit points your enemies have and the timing of the damage within the swing. Does it happen at the beginning? The end? Sometime else? (Give yourself partial credit if you recognize that once you're fighting enemies with more than about 50 hit points, these difference stop mattering and Sword A is reliably better.)

3. It's 78 degrees in Beaverton. Ninety miles away, in Salmon Falls, it's 72 degrees. Oakmont lies directly at the midpoint on a line between those two cities. What is the temperature there?

Answer: There is not enough information to answer this question. Ninety miles is too great a distance to assume a smooth linear transition, and you don't have a clue what terrain lies around Oakmont or how it generally affects regional weather patterns.

Suppose that last year there were 600 fatalities from car-related accidents. Also suppose that only 50% of all car occupants wore seat belts. How many fatalities will there be next year from car-related accidents if a mandatory seat belt law is passed?

Answer: There is not enough information to answer this question.

The biggest piece of information missing is how much safer a seat belt makes you. You can't get anywhere in solving this without that tidbit, and that's not something you can assume or even guess. It must be researched. But there's more you need to know to make a decent prediction. You need to know how many people would actually follow the law; you can safely assume that, in real life, it will not be 100%. You need to account for how many fatalities are pedestrians and bicyclists and motorcyclists, since they die in car-related accidents too but seat belts won't help them. You need to predict how many more or fewer cars will be on the road next year, and whether people will drive them more or less. Finally, and most insidiously, you need to predict if people's driving habits will change. It turns out that it's probably not safe to assume each person will get into the same number of accidents when everyone wears seatbelts as when only some people do, even if all other factors stay the same. Now, this rapidly gets into the realm of speculation and heated political debate, but there are states (Hawaii in particular) that saw an increase in death count per 100 miles driven after passing seat belt laws.

The weather report says there's a 40% chance of rain tomorrow at noon, 45% at 3:00 pm and 50% at 6:00 pm. What is the chance that it rains tomorrow?

I'll be honest: I have no idea. I threw this one in to get an answer from someone. It's not part of the test. I only know it's not 100% – (60% × 55% × 50%). You can only use that formula for independent events. Whether it rains at 3:00 pm is highly dependent on whether it rains at noon, and likewise for the 6:00 case.