Theoretical Probability

$$\textbf{Theoretical Probability}$$

Theoretical probability is just what it sounds like: the probability of an event occurring in theory. In other words, when you calculate theoretical probability, you are calculating a hypothetical or expected value. You are not calculating probability based on experimentally collected data (that would be Experimental Probability rather than Theoretical Probability).

We can calculate theoretical probability of an event, $E,$ occurring using the following formula:

Here, $P(E)$ represents the theoretical probability that event $E$ will happen, $n(E)$ represents the number of times $E$ can occur, and $n(S)$ represents the total number of outcomes.

Let's use this formula in an example!

We can tackle this problem in steps. First, let's figure out $n(S),$ the number of possible outcomes. Since there are ten marbles in the bag, the total number of marbles that you are going to be pulling from is ten. Thus $n(S) = 10.$

Now, let's find $n(E),$ the number of favorable outcomes. $n(E)$ represents the number of pink marbles there are in the bag. To calculate the number of pink marbles, we simply subtract the number of blue, purple, and green marbles from the total number of marbles:

There are a total number of $10$ marbles. Half of the marbles are blue. Thus $\frac{10}{2} = 5$ marbles are blue. We know that $2$ marbles are purple and $1$ marble is green. So,

Now that we have calculated $n(E)$ and $n(S),$ we can substitute this information into our theoretical probability formula!

$$\begin{align} P(E) &= \frac{n(E)}{n(S)}\\ &= \frac{2}{10}\\ &= \frac{1}{5} \end{align}$$

Thus, the theoretical probability of choosing a pink marble from the bag is $\frac{1}{5}.$

We already know that $n(S),$ the total number of marbles, is $10.$ Since there is only one green marble, we know that $n(E) = 1.$ Thus,

$$\begin{align} P(E) &= \frac{n(E)}{n(S)}\\ &= \frac{1}{10} \end{align}$$

Thus, the theoretical probability of choosing a green marble is $\frac{1}{10}.$

It is important to note that, if you drew a marble out of the bag, made a note of what color it was, put it back into the bag, shuffled the bag, and repeated this process multiple times, the actual amount of times you choose a certain color may not correspond to the theoretical probability of pulling that color. For example, you may pull a marble out of the bag five times, and you could pull out a green marble all five times. Theoretical probability does not tell you what will definitely happen in practice $-$ it just gives you an expected value.

What is a theoretical probability?

Think of theoretical probability as a "probability on paper." It's not something you find during an experiment. You aren't looking at data or thinking about what you've observed.

To say it another way, theoretical probability is just a hypothetical value. It's not based on what has happened, but rather on what should happen if you, hypothetically, decide to do an experiment or collect data.

There's a pretty straightforward formula we can use to calculate theoretical probability.

Don't be alarmed by the parentheses and letters in the equation! I wanted to include the "fancy" math equation you might see along with the translation into regular English.

$P(E)$ is a common abbreviation for the probability of an event, E, occurring.

So, to find the theoretical probability of an event, P(E), just divide the number of times the event E could occur by the number of all possible outcomes.

Let's practice.

How'd we get that answer? Let's take a look:

To find the total number of possible outcomes, we just counted up how many colors/sections there were on our spinner wheel. There are $6$ possible colors the spinner could land on, so our denominator is $6.$

(Note that the denominator ("total # of possible outcomes") includes green, which is the outcome we're interested in. Also note that all of the segments in our wheel are the SAME size. This means that landing on each color is EQUALLY LIKELY.)

Because all of the segments on our wheel are the SAME size, the theoretical probability of landing on one color is the SAME as the probability of landing on another.

We can say:

$P\left(\text{red}\right)=P\left(\text{orange}\right)=P\left(\text{yellow}\right)=P\left(\text{green}\right)=P\left(\text{blue}\right)=P\left(\text{purple}\right)$

Let's try another, slightly more difficult situation.

Note that all of our sections are still the same size, but we no longer have the same number of each color. This means the probability of landing each color will be different.

We live in a world which is incredibly random. Things we take for granted, like the time it takes to drive places, or the number of people in line for food, are not fixed permanently forever. They fluctuate based on unknown factors.

You use probability every day. For example, when you decide how much sleep you need, you are making a judgment about how likely it is that being tired the next day will affect you. (Admittedly, sometimes you will look over at the clock and wonder why it says $3\text{am}$ instead of $11\text{pm}$, but more often than not, we choose to pull an all nighter to finish a project or play games with friends till the dark hours of the morning because it's worth more to us than the problems we'll have tomorrow.)

When you make decisions like that, you are evaluating the Theoretical Probability that you will be tired tomorrow. Theoretical probabilities represent the hypothetical probability of an event occurring, but not necessarily the probability found by looking at any one set of those events which have already happened.

Theoretical probabilities are, in a certain sense, unknowable. There is no way you can know them for certain by experimentation alone. You can only find out by thinking about them in a probabilisitic way that makes assumptions about the way the world works.

This makes them different from Experimental Probabilities, which are found by doing experiments and are approximations of the "real" theoretical probabilities.

On the other hand, much of probability and statistics is based on the fact that if you do things a lot of times, then your results will eventually become so close to the theoretical probability that you can't tell the difference.

Let's take the unassuming example of a six-sided die. If you roll this die once, the probability of rolling a six is $\frac{1}{6}.$ Yet, when you roll a bunch of dice at one time $-$ let's say $30$ dice $-$ the number of sixes you roll is not necessarily going to be exactly $\frac{1}{6}$ of the total. In other words, you may not roll exactly $\frac{30}{6} = 5$ sixes.

Why is this?

Because each event happens randomly. If you rolled $6,000$ dice, it is unlikely that you would roll exactly $1,000$ sixes, but you would probably roll somewhere in that neighborhood. Of course, it is possible that you roll $0$ sixes, and $6,000$ ones, but this is highly unlikely. Thus, we can expect around $\frac{1}{6}$ of our rolls to be sixes, but there are also a range of other possibilities that can occur.

We assume (and are vindicated by experimentation) that on a fair $n$-sided die, the probability of getting any side is equal, and thus the theoretical probability of getting any one side is $\frac{1}{\text{Number of Sides}} = \frac{1}{n}.$ However, as discussed before, the theoretical probability may not always match up with the experimental outcome.

(Video) Experimental vs. Theoretical Probability

by Nat Oliver

Nat Oliver has a great video explaining what the difference is between theoretical AND experimental probability. He gives examples and explains how to calculate both.