### Video Transcript

Let π₯ denote a discrete random variable that can take the values zero, one, two, and three. Given that the probability π₯ equals zero is equal to one-ninth, the probability π₯ is equal to one is equal to four-ninths, the probability π₯ is equal to two is equal to π, and the probability π₯ is equal to three equals three π, find the value of π.

In order to answer this question, we will recall some of the key properties of discrete random variables. A discrete random variable has a countable number of possible values. The probability of each outcome can be described using a probability distribution function. If π of π₯ tells us the probability of each event occurring, then the sum of all π of π₯ values must be equal to one. And Each individual π of π₯ value must be between zero and one inclusive. And in fact, we can use this first property to answer the problem.

Weβre told that the probability π₯ is equal to zero is equal to one-ninth, the probability π₯ is equal to one is four-ninths, the probability π₯ is equal to two is π, and the probability π₯ equals three is equal to three π. And weβre also told that these are the only possible outcomes. So the probability that π₯ equals zero, one, two, and three, the sum of these is equal to one. We can replace each of these expressions with its corresponding probability value. So one-ninth plus four-ninths plus π plus three π equals one. One-ninths plus four-ninths is five-ninths. And π plus three π is four π.

We then solve this equation for π by subtracting five-ninths from both sides, giving us four π equals four-ninths. And finally, weβre going to divide through by four. Well, it follows that four-ninths divided by four is equal to one-ninths. And so given the information about our discrete random variable π₯, π must be equal to one-ninth.