Expected Number of Red Lights and Probabilities

Coming home from work, you always seem to hit every single light. Throughout time, you calculate the odds of making it through any light to be 0.2. On any given night, how many lights can you expect to hit before finally making it through one, and with what standard deviation? Finally, what's the probability of the third light you come across being the first one that is green?

The problem at hand is rooted in the concepts of geometric distributions and random variables. When dealing with problems of this nature, the main goal is to understand the probability distribution that best suits the scenario. Here, we're interested in two main aspects: the expected number of stops at red lights before getting a green light, and the probability that a specific sequence of lights will result in a green light at the third try.

We use the concept of a geometric distribution, which is particularly useful for modeling the number of trials needed for the first success in a sequence of independent and identically distributed Bernoulli trials, such as passing through a traffic light where each trial (light) can either be a "success" (green) or a "failure" (red).