Probability > Theoretical Probability

## What is Theoretical Probability?

The study of probability can be divided into two areas:

*Theoretical Probability*is the**theory**behind probability.*Experimental (empirical) probability*is probability calculated during**experiments**, direct observation, experience, or practice.

With theoretical probability, you don’t actually conduct an experiment (i.e. roll a die or conduct a survey). Instead, you use your knowledge about a situation, some logical reasoning, and/or known formula to calculate the probability of an event happening. It can be written as the **ratio of the number of favorable events divided by the number of possible events. **For example, if you have two raffle tickets and 100 tickets were sold:

Number of favorable outcomes: 2

Number of possible outcomes: 100

Ratio = number of favorable outcomes / number of possible outcomes = 2/100 = .5.

A **theoretical probability distribution** is a known distribution like the normal distribution, gamma distribution, or one of dozens of other theoretical distributions.

## Theoretical Probability Example

**Sample question:** What is the theoretical probability of rolling a 4 or a 7 with a set of two dice?

If this question asked you the empirical probability, you could set up an experiment. For example, you could roll the die a hundred times, record the results and state the probability. But as this question is asking you the *theoretical probability,* you need to use a formula or set up a sample space. As there is no single formula for calculating die rolling probabilities, set up a sample space.

Step 1: Set up a sample space. In other words, write out all of the possible “events” that can happen. In this case, the events are the numbers that come up after the dice are rolled. For two dice, the probabilities are:

[1][1], [1][2], **[1][3],** [1][4], [1][5], **[1][6],**

[2][1],** [2][2],** [2][3], [2][4],**[2][5]**, [2][6],

**[3][1],** [3][2], [3][3], **[3][4],** [3][5], [3][6],

[4][1], [4][2], **[4][3],** [4][4], [4][5], [4][6],

[5][1], **[5][2],** [5][3], [5][4], [5][5], [5][6],**
[6][1],** [6][2], [6][3], [6][4], [6][5], [6][6].

I’ve bolded the rolls that result in a total of 7.

Step 2: Figure out the probability. The entire sample space is made up of 36 possible rolls. There are 9 rolls that result in a 7, so the answer is:

9/36 = .25.

## Why not just Conduct Experiments all the Time?

There are several reasons why the field of Theoretical Probability exists. Sometimes, conducting an experiment isn’t possible for practical or financial reasons. For example, you might be studying a rare genetic trait in salamanders and you want to know what the probability of any one salamander having the rare trait is. If you don’t have access to all of the salamanders on the planet, you won’t be able to conduct an experiment so you’ll have to rely on theory to give you the answer. Theoretical probability is also used in many areas of science where direct experimentation isn’t possible. For example, probabilities involving subatomic particles or abstract structures like vector spaces.

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