Learn by Doing – Binomial Probabilities: Formula for P(X = x)
The purpose of this activity is to give guided practice at solving problems involving binomial random variables and to teach how the same probabilities can be found using an online calculator.
A multiple choice test has 10 questions, each with 5 possible answers, only one of which is correct. A student who did not study is absolutely clueless, and therefore uses an independent random guess to answer each of the 10 questions.
Let X be the number of questions the student gets right.
Give the formula expression for the probability distribution of X. In other words, apply the general formula for the probability distribution of a binomial random variable to the case in which n = 10 and p = 0.2.
What is the probability that the student gets exactly 4 questions right, P(X = 4)?
Applying the binomial formula is a good way for “first-timers” to understand the mechanics of binomial probabilities. Once you have mastered the technique, however, it may still be tedious to perform the necessary calculations.
For example, if I would ask you: What is the probability that the student will get at most 4 questions right? Or in other words, if we wanted to find P(X ≤ 4), we would need to add P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
For each of these 5 probabilities, we would need to use the formula, and then add the probabilities. This is very tedious. When calculations involve large n values, calculations become tedious as well. Luckily, any statistical software will do binomial calculations for us. In our course, we will use an online calculator for this purpose.
As practice, follow these steps to find P(X = 4) for our example (where n = 10 and p = 0.2), and verify that you get the same answer as you did in the last question, where you did it “by hand.”
Now use the online calculator to find the probability that the student gets no more than 4 questions right: P(X ≤ 4).
Use the online calculator to find the probability that the student gets more than 2 questions right, P(X > 2).
This document is linked from Binomial Random Variables.