### Video Transcript

Let ๐ be the random variable that represents the number of patients who visit a dental clinic per hour. The probability distribution of ๐ is shown in the table below. Find the probability of the following. (a) Exactly 13 patients visiting the clinic in a given hour. (b) At least 13 patients visiting the clinic in a given hour. (c) At most 13 patients visiting the clinic in a given hour.

So, weโre told that ๐ is the random variable. And it is a discrete random variable representing the number of patients who visit this dental clinic in any given hour. Weโre given the probability distribution of ๐ in a table. The probability distribution of a discrete random variable is the set of all values the variable can take together with their associated probabilities, which we refer to as ๐ of ๐ฅ ๐. Or we can write the probability that ๐ is equal to each ๐ฅ ๐ value. So, for example, the probability that ๐ is equal to 15, thatโs the probability that 15 patients visit the clinic in any given hour, is equal to 0.2.

Letโs consider the three parts of this question then. The first part asks us to find the probability that exactly 13 patients visit the clinic in any given hour. We can read this directly from the table. We find the value 13 in the top row. And then the probability associated with this is the value in the second row. Itโs 0.05.

In part (b) of the question, we were asked to find the probability that at least 13 patients visit the clinic in any given hour. This time, weโre going to need to find the sum of probabilities. At least 13 patients visiting the clinic means there could be 13, there could be 14, or there could be 15. So the probability that ๐ is greater than or equal to 13 is the probability that ๐ equals 13 plus the probability that ๐ equals 14 plus the probability that ๐ equals 15. We therefore need to sum the values 0.05, 0.3, and 0.2, which is 0.55.

Letโs now consider the final part of the question, in which weโre asked to find the probability that at most 13 patients visit the clinic in any given hour. This time, weโre looking to find the probability that ๐ is less than or equal to 13. And itโs less than or equal to because the question says at most 13 patients, so no more than 13 but 13 is okay. In the table then, there could be 13 patients or 12 or 11, or there could be 10. We therefore need to find the sum of these four probabilities, 0.2 plus 0.1 plus 0.15 plus 0.05, which is 0.5.

So weโve completed the problem. The probability that exactly 13 patients visit the clinic in a given hour is 0.05. The probability that at least 13 patients visit the clinic is 0.55. And the probability that at most 13 patients visit the clinic is 0.5.