Using Data to Predict Vaping Usage

Data from a National Institute on Drug Abuse study recently showed that in 2019, 25% of all American high schoolers used e-cigarettes. If we assume that this rate holds true at Hotchkiss, we can demonstrate some interesting conclusions using probability and statistics.

First, let’s introduce some concepts to help us with our calculation. A random variable is defined as a variable whose outcomes depend on some random phenomenon. For example, if X is the random variable describing the outcomes of a coin flip, X is heads with probability 0.5 and tails with probability 0.5. Another example is a dice roll. If we let Y be the random variable describing the outcomes of rolling a standard six-sided die, then the outcomes of Y are 1, 2, 3, 4, 5, and 6, each with 1/6 probability.

We can calculate the mean, or the expected value, of a random variable by multiplying the value of each outcome by its probability of occurring and adding all of these products together. We can denote the expected value of a random variable X with E(X). For example, if Y is the random variable describing the outcomes of rolling a standard six-sided die, then

For certain random variables, we can define a probability distribution function, or pdf for short. Basically, the pdf gives the probability that a certain outcome occurs. For example, if we have a random variable X with pdf f_X and x is one outcome of X, then f_X (x) is the probability that the outcome x occurs. Going back to our dice example, the probability of rolling a 2 is 1/6, so f_Y (2) = 1/6.

Now we’ll introduce a certain type of random variable - a Bernoulli random variable. A Bernoulli random variable has two possible values: 1, with probability p, and 0, with probability 1-p. We can model a coin flip with a Bernoulli random variable, for example. This is also perfect for modeling a Hotchkiss student - if a student vapes (with probability 0.25), then the outcome is 1; if not (with probability 1 - 0.25 = 0.75), then the outcome is 0.

However, what if we want to model more than one person? We can then use a binomial random variable. A binomial random variable models the number of trials that succeed when there are n trials and each trial has a p chance of succeeding. For example, we can use a binomial random variable to model the number of heads we get when we flip a coin n times. Thus, we can use model the number of students who vape using a binomial random variable.

There are a few useful facts about binomial random variables. If X is a binomial random variable with n trials and each trial has a p chance of succeeding, then E(X) = np. Furthermore, the pdf of X will be

.

We can let X be the random variable that describes the number of students at Hotchkiss who vape. For the purposes of this article, let’s assume that there are 600 students at Hotchkiss. Then we can model X as a binomial random variable with n = 600 and p = 0.25.

We can immediately see that the expected value of X, E(X), is equal to 600 • 0.25 = 150 students. This means that the expected number of students who vape at Hotchkiss is 150.

We can also calculate the probability that certain numbers of students vape in certain groups (classes, dorm floors, friend groups, etc.). We can do this by summing up the various probabilities that a certain number of students vape. The TI-nspire’s binomCdf function makes this easy by adding up all the different pdfs for us. For example, the probability that at least that one person in a group of two people vapes is 0.4375: