 It's helpful to introduce the concept of a random variable. For our purposes, introductory probability, we define a random variable as follows. Given a sample space S, a random variable X is a function that assigns a unique real number to every element of S. And so the range of the random variable corresponds to the elements of our sample space. So, for example, let X be a random variable for the experiment, roll a 3-sided die, and record the outcome. And since our random variable is a function, there are many possibilities, so let's just find one of them. So the sample space S consists of the possible outcomes when a 3-sided die is rolled. So what we'll need to do is determine what real number we'll assign if the die shows 1, 2, or 3. And so we might have 0 if the die shows 1, pi if the die shows 2, and 0, how about square root of 7 divided by 1 plus the 5th root of e if the die shows 3. Actually, we'll probably use 1 if the die shows 1, 2 if the die shows 2, and 3 if the die shows 3. Now, the real number we assign isn't always that obvious, so for example, let Y be a random variable for the experiment, a letter is drawn from a bag of letter tiles. And again, Y is a random variable, which is a function, and so there's an infinite number of possibilities, let's just find one of them. So the sample space consists of the letters A through C, so we need to know what to do if the tile is A, B, C, and so on down to Z. And we might make the following choice. We could let Y be 1 if the tile is A, 2 if the tile is B, 3 if the tile is C, and so on. Now, there's really two types of random variables. A discrete random variable is one that can take on specific values in an interval. For example, the number of heads in 10 flips of a coin, the number of boy-children in a family. Meanwhile, a continuous random variable is one that can take on all values in an interval, or perhaps even multiple intervals. For example, the height of a tree, or the weight of a car. Now we've already found probability distributions for discrete random variables, but what about continuous random variable? Now before we do that, there is an important problem we have to address. Suppose we have a continuous random variable. What's the probability a data value is exactly equal to a given number? For example, a person's weight is exactly 135 pounds. And by exactly, we mean exactly, not 134.999 pounds, and not 135.0000001 pounds. Or again, perhaps we might ask about the probability the height of a tree is exactly 2.5 meters, and not a trillionth of a meter more. And intuitively, we would not expect any continuous quantity to have a value exactly equal to a given number, and so the probability should be zero. To resolve this, we note the following. When we claim a person's weight is 135 pounds, we really mean that our scale reports it as 135 pounds. But that report is limited by the accuracy of the scale, and so we can think of this measurement in accuracy as a result of round off. And so when we say 135 pounds, what we really mean is any value between 134.5 and 135.5 pounds, because this is the interval that will round to 135 pounds. And so when we talk about the probability, the probability is really the probability of the weight is between 134.5 and 135.5. For example, consider the experiment where a coin is flipped 100 times and the number of heads is recorded. What values of a continuous random variable x correspond to the event that 60 heads are obtained? And the important idea here is we look for an interval that will round to 60. So the smallest number that rounds up to 60 is 59.5. And all higher numbers will round to 60 until we get to 60.5. And so the event 60 heads corresponds to the interval 59.5 less than or equal to x less than 60.5. Now as is often the case, the hardest part of this is interpreting the English language. So consider the experiment where the number of defects in a 100-meter coil of wire is counted. What values of a continuous random variable y correspond to the event that at least five defects are found? So again, we want to find the smallest number that rounds up to at least five, and that will be 4.5. So anything greater than or equal to 4.5 will round to at least five, and so the interval will be y greater than or equal to 4.5. Now it turns out it will be convenient to always have the lower number on the left side of an inequality. And so it's more convenient for us if we were to write our interval as 4.5 less than or equal to y.