So this is a problem I always thought about ever since I was 7 years old and had recently learned about the most basic probability (coin probability). A light changes once every two minutes. A driver pulls up to the light with no idea as to when the light last changed. However, he knows that the light changes periodically every two minutes. The driver is then faced with a tough question: How long should he keep his engine running to minimize the expected amount of gasoline used?

Car specs:

While idle with the engine on, his car uses .5 gallons of gas per minute (gpm).

If he turns the engine off, it requires .4 gallons of gas to turn the engine back on.

I feel like the answer is that he should either immediately turn his engine off or keep it running depending upon which one is greater: gpm or gallons used to restart the car. However, are there any values of gpm and gallons used to restart that yield an answer where the man has to keep his engine on for a little and then turn it off? I think the only scenario where this happens is when the probability of the light turning green is not the same at all times, but rather has a probability of changing that is proportional to time. What do you guys think?

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