He clearly didn't give it 110%

November 17, 2009 in Math

Silicon Alley Insider is running a series of posts called "15 _______ questions that will make you feel stupid." The blank has been filled twice with "Google interview" and most recently with "management consultant interview."  I particularly enjoyed one of the Google questions:

If the probability of observing a car in 30 minutes on a highway is 0.95, what is the probability of observing a car in 10 minutes (assuming constant default probability)?

I have no idea how the word "default" snuck in there - I'm guessing whoever wrote this had a need to relate things back to dangerous CDS! - but the question is a good one. However, the answers posted on the site are absolutely horrendous. One ardent commentator wrote:

"observing a car in 30 minutes on a highway" 
If 30 min = 95%, then 100% probability = 30/0.95 => 31.5 min 
(ie, the max interval between 2 cars could be 31.5 min) 
Probability in 10 min = 10/31.5

You have to wonder if, by his logic, there's really a 110% chance of seeing a car in 34.7 minutes?

The correct answer is below...

  1. The probability of observing no cars in 30 minutes is 1-95%, or 5%
  2. The probability of observing no cars in 10 minutes, p, must agree with the statement p^3 = 5%, since three consecutive carless 10 minute periods will pass with 5% probability.
  3. Therefore, p = 36.8%
  4. And the probability of observing a car in 10 minutes is 1-p, or 63.2%.

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