# Discrete Probability Distributions

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- Understand what a probability distribution is and what it means
- Be able to differentiate discrete outcomes from continuous ones
- Know what properties are unique to discrete distributions
- Understand how discrete distributions may have a finite or infinite number of outcomes

As we become more familiar with probability, it becomes necessary to work with specific types of probability distributions. This lesson introduces discrete distributions (separate countable outcomes), and specifies several properties that they must have to be legitimate. We'll also see both tabular and function forms of defining the potential outcomes and their respective probabilities.

Practice problems and worksheet coming soon!

## Theoretical Distributions

With a firm understanding of what probability is, we can begin to study probability distributions. Up until now, we've worked with probabilities that were either pre-defined or success/failure based, such as the probability of flipping heads on a fair coin, or being told that the probability that Johnny gets an A in history this year is $75\%$. Probability distributions allow us to look at events with multiple outcomes.- The probability of each outcome is between $0$ and $1$, where $0$ means the outcome is impossible and $1$ means the outcome is certain to happen
- The sum of all probabilities in a distribution must equal exactly $1$, so that the distribution contains every possibile outcome, since a sum to $1$ captures all event outcomes

## Telling Apart Discrete from Continuous

We recently learned that random variables » can be either discrete or continuous. Discrete random variables are ones in which the outcomes are countable, while continuous random variables are ones for which it makes sense to have partial values for outcomes. Probability distributions exist for both cases, but this lesson focuses on discrete outcomes. After all, it is only the discrete outcomes that we can make a list for, since they are countable. If you are working with a continuous random variable, like time, distance, or sometimes money, do not try to apply the table format of distributions that we work with in this lesson.- The random variable for the number of flowers in a randomly selected house's backyard
- The random variable for the number of coins a person can take in one grab out of a jar full of coins
- The random variable for the distance between two best friends at your school at any random time
- The random variable for amount of electricity a randomly selected house in your neighborhood used last month

- The random variable for the number of flowers in a random house's backyard
- This is a discrete situation because the outcome is countable, even though the number probably has a large variance and high expected number in many cases.
- The random variable for the number of coins a person can take in one grab out of a jar full of coins
- Even though this is related to money (which is often considered continuous due to partial values), this situation measures the number of coins grabbed, which is a countable outcome and therefore discrete.
- The random variable for the distance between two best friends at your school at any random time
- Distance is almost always a continuous measure because we do not use individual countable units of measure but rather measure in whatever unit, such as feet, meters, or yards, and allow for partial values, e.g. $15.4$ meters.
- The random variable for amount of electricity a randomly selected house in your neighborhood used last month
- Electricity is not measured in a countable way like the number of beans in a jar is. Electricity is a continuous measure because we can have partial values (whether or not you know exactly what units we measure amounts of electricity in, hopefully this is a familiar concept to you).

## Finite Outcome Distributions

While this lesson focuses only on discrete situations, we can still have a finite or an infinite number of outcomes. Fortunately, having a finite number of outcomes is by far the most common case, which means we can jot down every outcome in table form alongside the probability that it happens. This is necessarily the case when we work with distributions that have non-number outcomes, such as we'll see in Example 2.Every valid probability distribution has the feature that the sum of all outcome probabilities is $1$. If this is not the case, then the distribution is either invalid or missing outcomes from its list. Make sure to double check that this is the case.Table form is very convenient for several types of common calculations. If a finite outcome discrete situation is described to us in words, we may need or want to turn it into tabular form.## Function-Defined Distributions

While we are most commonly given a distribution in table form or a word description, it is also possible to use function notation to describe distributions. We can get any individual outcome probability by plugging in the $x$ value. If there are few enough possibilities, we can also use the function to generate a table.## Infinite Outcome Distributions

While they are not nearly as common, it is likely that you will encounter some discrete distributions with an infinite list of possibilities. These distributions will always have a formula-based approach for calculating outcome probability, and typically have special names. One of the most common ones is called the Poisson Distribution » and works well for measuring outcomes that involve rates.We'll save the deep-dive of the Poisson distribution for its own lesson (if you even need to learn it - it's not on the AP Stats syllabus), but leave you with a quick example of what the formula and distribution may look like in context.## Probabilities of Ranges of Outcomes

In order to quickly calculate commonly requested quantities, mathematicians created two types of functions related to probability outcomes of discrete distributions - 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 of attaining a specific outcome or fewer. You probably don't need to memorize what the acronyms stand for, but you should have a strong understanding of each of these objects.## The Complement Rule

When we have to work with table forms of distributions, or in other cases that require by-hand calculations, we can leverage a shortcut called "the complement rule of probability".Consider the following distribution for the random variable $X$:$$ \begin{array}{|c|c|} \hline x & \mathrm{Pr}(X=x) \\ \hline 0 & 0.04 \\ \hline 1 & 0.07 \\ \hline 2 & 0.13 \\ \hline 3 & 0.16 \\ \hline 4 & 0.20 \\ \hline 5 & 0.15 \\ \hline 6 & 0.10 \\ \hline 7 & 0.06 \\ \hline 8 & 0.03 \\ \hline 9 & 0.01 \\ \hline 10 & 0.04 \\ \hline 11 & 0.10 \\ \hline 12 & 0.06 \\ \hline 13 & 0.03 \\ \hline 14 & 0.01 \\ \hline 15 & 0.04 \\ \hline \end{array} $$What if we were asked for the CDF of 13? It would be tremendously more difficult to get it by adding up the cumulative probabilities for outcomes zero through thirteen than it would if we remembered that all probabilities add to $1$, and so we can start with $1$ and subtract away the outcomes that we don't want. This shortcut is called the complement rule because in probability, the complement of any probability is $1$ minus that probability. If the probability represents the likelihood that the outcome will happen, the complement represents the probability that the outcome won't happen.$$\mathrm{CDF}(13) = 1 - \mathrm{Pr}(X=14) - \mathrm{Pr}(X=15)$$$$=1 - 0.01 - 0.04 = 0.95$$## Mr. Math Makes It Mean

Conditional ProbabilityConditional probability » (sometimes called dependent probability) is among the more confusing concepts in probability. We'll cover it in more detail a few lessons from now, but it's possible you may have already seen it in your classwork by the time you are beginning to learn the ins and outs of discrete distributions.In short, conditional probability requires a special formula to adjust for questions in which we know some bonus information about an outcome. If we know event $B$ happened, the probability that event $A$ happened may not be the same as if we didn't know that event $B$ happened. The formula is$$\mathrm{Pr}(A | B) = \frac{\mathrm{Pr}(A \cap B)}{\mathrm{Pr}(B)}$$The biggest challenge in using conditional probability with any random discrete distribution we may be working with is simply putting all the knowledge and details together correctly. Make sure to look at the dedicated lesson » if you are new to conditional probability, and feel free to skip this section for the time being if your class hasn't got there yet (though learning by example in this lesson will be very helpful!).## Put It To The Test

Example 8For each of the following random variables, determine if it is continuous or discrete.$1)$ The amount of soup served in a restaurant on any given night$2)$ The number of pennies in a coin jar$3)$ The chair count at a randomly selected wedding hall$4)$ The amount of wood needed to build a house- Understand how distributions describe the likelihood of outcomes in any situation
- Know the core characteristics that must be true of any probability distribution
- Continue to develop an understanding of the difference between discrete and continuous situations
- Be able to write down your own table of outcomes and probabilities based on a word problem description
- Know when discrete outcomes need to include or not include zero, and when it makes sense for outcomes to theoretically be infinitely large even though such outcomes are unlikely to happen (see the bank example)

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