# The Binomial Distribution

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Lesson Priority: VIP Knowledge

- Know the specific circumstances that define a binomial distribution
- Understand what probability the binomial distribution measures
- Know and memorize the formulas for the mean and the standard deviation of a binomial distribution
- Master the difference between the "pdf" and the "cdf"
- Be able to get fast answers from your TI-84 Calculator for binomial problems

Due to its simplicity and far-reaching applicability, the Binomial Distribution is one of the most common discrete distributions. It quantifies how likely it is to have a certain number of successes in $n$ attempts, given a fixed per-trial success probability. E.g. this distribution can tell you how likely it is that you got 10 out of 20 SAT questions correct if you were guessing blindly. This lesson will introduce both the concept of the binomial distribution as well as the formulas you will be expected to retain for quizzes and tests on this distribution.

Practice problems and worksheet coming soon!

## Measuring Success

The Binomial Distribution measures the probability that, out of $n$ identical attempts at a success / failure objective, $k$ of those attempts will be successes.In order to use this distribution correctly, we need a few things. First, we need to understand when this concept does and does not apply. Second, we need to understand what we're measuring and what the correct interpretation of the probabilities and expected values is. Finally, we need to be able to get these results practically, which depends entirely on how much your teacher allows you to depend on Calculator Magic.## When Can I Binomial?

As we said, the probabilities we calculate with the Binomial Distribution are for success / failure situations. Specifically, Binomial random variables measure the number of successes only for situations involving repetitive trials of the same objective. Because it is the same repetitive objective, the Binomial Distribution formulas require that the probability of each trial being a success must be the same. This is a very important requirement that trick questions will test you on!We also need to be able to assume that successive trials do not affect one another - in other words, that they are independent.Here are some examples of situations that we can measure with the Binomial Distribution:- The probability for the number of times we get an even roll when we roll a fair die 10 times
- The probability for the number of correct multiple choice SAT questions a zoo animal answers correctly out of 20
- The probability for how many times a particular artist's song shows up during an hour of internet radio

- The probability for how many times out of ten dice rolls that you roll a "four", where each time you use a different die, using some die that are six-sided and some that have more sides
- The fact that each die roll is using a different die, and that each die may or may not be a traditional six-sided die, ruins the required assumption that each trial has the same success probability

- The probability for hitting a certain number of bulls-eyes thrown by a dart thrower who is throwing 100 darts in a row
- This isn't a situation where random chance is the only factor, and it is probably not true that this dart thrower has the same chance of success of hitting a bulls-eye on each attempt - reasonably, successive throwing may tire him out, making later throws less likely to succeed.

## What Are We Quantifying?

In general, a probability distribution gives you a probability associated with a specific outcome. It is both helpful and necessary to have a strong understanding of what you're looking at when you get a result from a binomial situation.In any binomial situation you're examining, there are $n$ identical trials being done repetitively, with each trial having a probability of $p$ of being successful. The probabilities we get as answers tell us the likelihood of $k$ number of trials being successful, where $k$ is a number smaller than $n$.Follow along with the descriptions, probabilities, and notations below to put this concept into context.## The Binomial Formula

The fortunate among you, those who know their class is going to 100% rely on a TI-84 or similar calculator, can likely read this section as bonus knowledge. While you do need some sort of calculation to carry out the formula, the formula is quickly becoming "the old" way of doing this, since TI-84 type calculators do the formula for you.## Binomial Mean

We need to know what mean or expected value that a binomial distribution has, which is something we'll often want to know with any probability distribution. Unlike a "straight average" which you probably know how to do with some given values, the mean of a binomial distribution represents the average outcome we would expect if the entire experiment was conducted millions of times.For context, in our $15\%$ probability $10$-draw M&M example, we found that the probability of $X$ (the number of blue M&M's that were about to be drawn) being $3$ was about $13\%$. We could have done this for any of the other possible outcomes, $0$ through $10$, but that still doesn't answer the question - what would happen if we did this entire $10$-draw experiment millions of billions of times? What is the "average" outcome?There are two ways to proceed. In an earlier lesson », we learned that for any probability distribution, the mean, also known as the "expected value", can be obtained by adding the products of the values and the probabilities. For example, here is the full probability distribution for the blue M&M example.$$\begin{array}{|c|c|} \hline x & \mathrm{Pr}(X=x) \\ \hline 0 & 0.197\\ \hline 1 & 0.347\\ \hline 2 & 0.276\\ \hline 3 & 0.130\\ \hline 4 & 0.040\\ \hline 5 & 0.009\\ \hline 6 & 0.001\\ \hline 7 & 0.000\\ \hline 8 & 0.000\\ \hline 9 & 0.000\\ \hline 10 & 0.000\\ \hline \end{array}$$Probabilities are rounded to the nearest $0.001$, so while the likelihood of $7$ or more blue M&M's rounds to $0$, it is not actually equal to $0$ (but these outcomes are very, very close to zero because they are very, very unlikely to happen). The probabilities sum to $1$ as expected, since a probability of $1$ encompasses all possibilities.You can use this table to get the expected value of this distribution by adding the products of the values and their corresponding probabilities.$$\mu = (0)\cdot(0.197) + (1)\cdot(0.347)$$$$ + (2)\cdot(0.276) + \dots + (9)\cdot(0.000) + (10)\cdot(0.000)$$$$\approx 1.500$$Alternatively, there is a one-shot quick formula that will give you the expected value of a binomial distribution. It's a common theme in probability that "named" distributions, ones that have a name and formulaic probability approach, typically also have formulaic expected values.## Binomial Variance and SD

The discussion about the variance and standard deviation of this distribution is nearly identical to the one we just had about expected value, in that there is a way to do it "the long way" using generic approaches we learned for probability distributions that work for the binomial distribution because they work for any distribution.However, the variance of a binomial random variable has a simpler formula, and we are expected to know it.## PDF Versus CDF

Recall from the introductory lesson on Discrete Probability Distributions » that we work with two types of probability results: the PDF and the CDF. The PDF (probability density function) describes the probability associated with a specific outcome, while the CDF (cumulative distribution function) describes the probability associated with the outcomes that are less than or equal to a certain amount of successes.Looking back at our M&M's situation, we would say that PDF($6$) represents the probability that we draw six blue candies, while CDF($6$) represents the probability that we draw six or fewer candies.This distinction is plain enough, but the challenge will come from the wording of questions. It may seem like a cheap shot, but textbooks and teachers are infamous for tricky questions that can have slightly different meaning depending on changing a single word.It is worth a reminder that there is no formula for a discrete CDF. Beside using a calculator, the only thing you can do is add up individual PDF results. For example, CDF($6$) = PDF($0$) + PDF($1$) + PDF($2$) + PDF($3$) + PDF($4$) + PDF($5$) + PDF($6$). Fortunately, we will almost always use a calculator. Additionally, there is a complement trick for computing CDFs that are on the high end of the range, which we will see below.We'll see how to use PDF and CDF to answer the right question, but first let's know our way around the calculator.## Using Your Calculator

Using the TI-83 / 84 calculators, we can get instant probabilities for both PDF and CDF situations.First, choose DIST (by pressing 2nd $\longrightarrow$ VARS)Select binompdf, which is option "0" or "A" depending on your model. Also depending on your model, you will either be brought to a screen like thisor this, if your calculator software is older:The first case has a nice interface - just literally tell it what your $n$, $p$, and desired $x$ is. For example, for a binomial situation with $7$ trials with each trial's success probability equal to $20\%$, then $n$ is $7$ and $p$ is $0.2$. The desired $x$ is whatever probability you're asking for. For example, if you want to know the probability that $3$ trials are successful in this situation, $x$ is $3$.If you have older software and you have to enter the numbers yourself, you have to remember 1) the order to enter them ($n$ then $p$ then $x$), and 2) to use the comma button , to separate each of the three numbers. When you're done, using either method, you should see something like this on your screen:Hit the enter button, and viola.The process for getting CDF probabilities from your TI is nearly identical. The only difference is you choose the option for "binomcdf" instead of "binompdf".## Answering the Right Question

The last thing you need to start answering quiz questions like a champ is to understand what question you're being asked. A good first step is to identify whether the probability they're asking for is a single outcome or a range of outcomes. Then, we should set up the appropriate PDF or CDF expression. The rest is done on your calculator (hopefully).Here is the binomial situation we will use to work through some clarifying examples.## Mr. Math Makes It Mean

Once you've mastered the ins and outs of binomial situations and probability calculations, teachers will often ask questions that implement conditional probability rules.Recall that the dependent probability formula is$$\mathrm{Pr}(A | B) = \frac{\mathrm{Pr}(A \cap B)}{\mathrm{Pr}(B)}$$Let's see an example of how this can be implemented in a binomial situation.## Put It To The Test

Give a few descriptive problems a try, and make sure to check out the practice worksheet for this lesson!- Understand binomial outcomes as groups of success / failure trials
- Know what assumptions need to be validated for a success / failure situation to actually be binomial
- Be able to set up word problems using PDF and CDF expressions
- Know how to use your calculator to get whatever binomial probabilities you seek
- Be well-practiced at the common types of questions you could be asked on a test, and the tricky word choices teachers like to use

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