## Lost Car Keys

A forum for good logic/math puzzles.

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DataGenetics
Posts: 24
Joined: Sat Jan 01, 2011 1:12 am UTC
Location: Seattle

### Lost Car Keys

I drive to work, and put my car keys in my pocket. Whenever I sit down in a meeting there is a 20% chance that my car keys might fall out of my pocket (unnoticed). At lunchtime, I reach for my keys and find out they are missing. If I’ve had three sit-down meetings that morning, what is the chance that the car keys fell out in the first meeting location?

Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

### Re: Lost Car Keys

Before checking your pocket, there's a 1/5 chance the keys fell out during the first meeting, a 4/5*1/5 = 4/25 chance the keys fell out during the second meeting, a 4/5*4/5*1/5 = 16/125 chance the keys fell out during the third meeting, and a 4/5*4/5*4/5 = 64/125 chance the keys are still in your pocket. After you check your pocket and find the keys not there, Bayes' Theorem tells us that the probability your keys fell out during the first meeting is the prior probability of that happening divided by the total probability the keys fell out of your pocket, or (1/5)/(1/5+4/25+16/125) = 25/61.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.