Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.
Understanding Discrete Variables
Brady is playing a game with his friends. He needs to roll a 4 to avoid any bad things from happening to him in the game. He has two chances to roll a 4. Before he rolls the die, his friends bet if he is going to make it.
''Never gonna happen,'' says one.
''He's got a 50% chance!'' says another.
''You're wrong!'' says a third friend.
So what are Brady's chances of rolling a 4? Are any of his friends right?
In this lesson, you'll figure out Brady's chances of rolling a 4 by understanding discrete variables and learning how to use a formula to find the expected value of a discrete variable. First, let's define discrete random variables.
A discrete variable is an outcome of discrete data, which is data that cannot be divided; it is distinct and can only occur in certain values. In other words, a discrete variable is an experimental result that cannot be divided. For example, if you were to count the number of people in a classroom you would have a discrete variable because you can only have a whole person, not a half or a quarter of a person.
Brady's variable is discrete because he can only roll whole numbers with the die. We can also say that Brady's variable is random because he has no control over how the die will land. Experiments such as drawing a card from a deck, flipping a coin, or rolling a die are all examples of discrete random variables.
Now, let's talk about how you can predict the probability of a discrete random variable using a concept called expected value.
This is a 6-sided die:
Each number (1, 2, 3, 4, 5, and 6) is represented on one side of the die. There is only one number on each side. These are important facts to keep in mind when finding the expected value of a discrete random variable. An expected value is simply the number of successful outcomes expected in an experiment. In Brady's case, the expected value is the probability that he will roll a 4 in two tries. Note that this is how you calculate probability or what you expect to happen, not necessarily the actual result of the experiment.
The formula for expected value of a discrete random variable is n * P. This is also considered the mean or average probability. The n represents the number of trials, and the P represents the probability of success on an individual trial. Before we solve Brady's problem, let's look at a different example to help you understand expected values.
Pretend there is a nice dinner happening and there is a room full of men and women. If you sit outside the room, what is the probability that the next person to walk out of the room will be a man? Well, if after you sit outside there are still ten men in the room and ten women, you can probably guess that you have a 50/50 chance that the next person to walk out of the room will be a man. 50% is the expected value in this scenario because it is the probability of what you expect to happen.
Going back to Brady's problem, we need to know n, or the number of trials, and P, or the probability of success of rolling a 4 in a single trial, before we can find the expected value. First, we know the number of trials is 2 because Brady has two chances to roll the die. We also know that the probability of rolling a 4 in a single trial is 1 out of 6 or 1/6 because there are 6 sides and one 4 on one of the sides. Now we can plug our numbers n = 2 and P = 1/6 into our formula: 2 * 1/6. Simply solve to find the expected value: 2 * 1/6 = 0.3333.
Therefore, the expected value of rolling a 6-sided die two times and getting a 4 at least once is 33%. Or you could say that, on average, when a die is rolled two times you get a '4' 33% of the time.
When working with discrete variables, it can be important to understand those variables and analyze them before an experiment is conducted. A discrete variable is an outcome of discrete data, which is data that cannot be divided; it is distinct and can only occur in certain values. Remember that in Brady's scenario, he couldn't have half of a 4 when rolling a die.
Today, we wanted to use discrete data to find the expected value, which is the number of successful outcomes expected in an experiment. To do this, we used the formula n * P, where the n represents the number of trials and the P represents the probability of success on an individual trial. To find the expected value, simply take the number of trials and multiply by the probability of success on an individual trial.
For more information about probability and discrete data, check out our other lessons!
After you have finished with this lesson, you should be able to:
- Define discrete data and expected value
- Identify the formula for finding the expected value with discrete data
- Explain what the variables are in this formula and how to calculate expected value
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