# Statistics traffic light problem.

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.

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

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 $$\displaystyle \sum_{k=0}^{5}P(X=k)$$