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Through several hundred trials, on any given commute on a particular street road, the probability that a taxi car needs to legally stop at a traffic light is approximately 45%. If all traffic lights in a city are independent of one another, what is the probability that a randomly selected taxi driver will be able to legally drive through at-most five traffic lights until they need to stop at the traffic light?

My guess: It is a binomial distribution? Not sure how to deal with the sample size or the 5 sequential successes.

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  • $\begingroup$ What is the chance the taxi can go through two lights without stopping? This is just multiplication-five successes followed by a failure. $\endgroup$ Commented Mar 23, 2020 at 4:11

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Let $X$ be the number of traffic lights that a randomly selected taxi driver will be able to legally drive through.
Observation: $P(X=k)=(1-0.45)^k(0.45)=0.55^k\cdot 0.45$ since the driver has to drive through the first $k$ traffic lights and stopped by the $k+1$th one.

Then the required probability is just $\displaystyle \sum_{k=0}^{5}P(X=k)$

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