Traffic Light
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
There is a traffic light at the top of a hill. Cars can't see the light until they are 200 feet from the light.
The cycle of the traffic light is 30 seconds green, 5 seconds yellow and 20 seconds red.
A car is traveling 45 miles per hour up the hill.
What is the probability that the light will be yellow when the driver first crests the hill and that if the driver continues through the intersection at her present speed that she will run a red light?
Answer
The probability of the driver encountering a yellow light and the light turning red before the car enters the intersection is about 5.5%.
At 45 mph the car is traveling at 66 feet/second and will take just over 3 seconds (3.03) to travel the 200 feet to the intersection. Any yellow light that is in the last 3.03 seconds of the light will cause the driver to run a red light.
The entire cycle of the light is 55 seconds. 3.03/55 = 5.5%.
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Comments
precious1026
Feb 19, 2009
 This is probably a good teaser. 
Zag24
Feb 25, 2009
 Of course, we all know that you speed up when you see a yellow light. 
opqpop
Sep 28, 2010
 Wasn't really sure what you meant by "crest the hill."
Other than that, decent teaser. 
zembobo
Jun 22, 2011
 This teaser is inaccurate in two ways. First off, you didn't ask for the total chance of running a red light but if you did your answer is still wrong because the light is already red for 20 of the 55 seconds. If it's red but going to change to green within 3.03 seconds she's ok so theres 16.97 seconds of red light danger + 3.03 yellow light danger which brings it back up to 20. 20/55=36% chance of running a red light.
But (secondly), the teaser actually asked the probably of running a red light if the car SAW a yellow at the crest and KEPT GOING. So we only need to worry about 5 seconds. Where along the 5 seconds of yellow light is she? 3.03 of that 5 is danger. 3.03/5=~61% FTW. My answer makes more common sense too because it's definitly greater than 50% if it's going to take her 3 out of 5 seconds just to get to the intersection and the light is already yellow. 
zembobo
Jun 22, 2011
 Ya know what though, I suppose you could read the puzzle differntly..like what are the chances she she's a yellow light AND runs a red light...(5/55)*(3.03/5) = 5.5%. I apologize if that's what you were asking. Perhaps there's more than one way to read/solve this thing. Got me thinking though! :) 
javaguru
Oct 13, 2011
 I think it read quite clearly: what is the probability of both the events occurring together?
The question didn't ask what the probability was *if* she saw the yellow light and then proceeded. 'If', 'and', and 'or' all have very welldefined meanings in the domain of probability. 
eighsse
Jul 27, 2013
 I got it wrong because I missed they very important word "AND." Also in some places, being within the intersection when the light turns red is considered running the red light. So you could add 40 or 50 feet to the 200 if it is in one of those places (though unusual). 
cms271828
Jun 07, 2014
 I got 200 / (66x55) = 5.50964%
You basically just work out what part of the time the orange light is on that allows the lights to change to red before you get there. Then divide it by 55. 
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