Expected value word problem from Khan Academy

Question

Hello all and thanks for taking the time to read this. I'm closing out the last few sections of statistics in Khan Academy, but there is a problem that is really bugging me. The problem reads like this. Marvin is supposed to buy a light for his moped, but he's trying to argue the case that it will be less expensive without it. A light for his moped will cost $150. He compiles accident stats into a table like so:

Time of Accident|Cost  |Probability
----------------|------|-----------
Morning         |$2000 | 10%        
----------------|------|-----------
Dusk            |$4000 | 15%
----------------|------|-----------
Night           |$2000 | 20%

It then goes on to say that it doesn't matter if he has the light during the day or dusk, but it will prevent him from having an accident at night. I would have gotten the problem wrong anyway because I thought the final value was going to be negative, but here was my thinking:

Expected cost of accident with purchase of moped light

[P(morning accident) * (cost of morning accident + cost of light)] + [P(dusk accident) * (cost of dusk accident + cost of light)] + [P(night accident) * (cost of night accident + cost of light)] 

Plug in the values like so:

[.1 * (2000 + 150)] + [.15 * (4000 + 150)] + [0 * (2000 + 150)]

So obviously I got this wrong, but I'm not sure why. Khan gave the answer as this:

[.1 * (2000)] + [.15 * (4000)] + [150]

I'm baffled by this! Isn't the probability of a night accident zero?! And why isn't the cost of the light factored into the other accidents? Can anyone help me out with this? Any help would be much appreciated! I'm including a screen shot, hopefully you can read it well enough.

EDIT Thanks to Matt Gunn for his post, I was able to work through the problem again. My problem was the fact that I did not account for the possibility of not getting into an accident. This should have been my thinking from the start.

[P(morning accident) * (cost of morning accident + cost of light)] + [P(dusk accident) * (cost of dusk accident + cost of light)] + [P(night accident) * (cost of night accident + cost of light)] + [P(NO ACCIDENT!) * (cost of no accident + cost of light)]

Plugging the values back in like so gives:

[.1 * (2000 + 150)] + [.15 * (4000 + 150)] + [0 * (2000 + 150)] + [.75 * (0 + 150)]

Which simplifies to:

(.1 * 2000) + (.15 * 4000) + [150(.1 + .15 +.75)]

And finally gives the answer of 950. So, the cost of the light was factored into the cost of an accident, I just wasn't able to see that Khan Academy had skipped the steps where that value was factored back out. Thanks again for the help!

Answer 1

In your formulation (adding the cost of the light to the cost of an accident), Marvin will only pay for the light if he gets in an accident... That's generally not how the purchase of vehicle parts work :P

You could either:

1. Do what they did (i.e. add 150 to the expected cost of an accident).
2. Do what you did but add an additional .75 * 150 to pay for the light in the 75 percent of the time you don't get an accident.

Of course, both (1) and (2) are equivalent since: $$.1*150 + .15*150 + .75*150 = (.1 + .15 + .75) * 150 = 150$$