*Contributor: Mason Smith. Lesson ID: 11309*

Theoretical probability can save you the trouble of repeating experiments over and over. Sound good? Then don't get out your die or coins, but work online and in your brain to figure the odds of odds!

categories

subject

Math

learning style

Visual

personality style

Lion

Grade Level

Middle School (6-8), High School (9-12)

Lesson Type

Quick Query

Q: What is the probability of a coin landing on heads 20 times out of 100?

A: A lot higher than the probability that I'll stand around to try it out!

*Theoretical probability* has often lovingly been referred to as "experimental probability without the work" by anyone who doesn't want to stand around flipping coins.

The main idea of theoretical probability is based on the idea of working smart rather than working hard. Instead of spending countless hours conducting thousands of experiments, we just use the "rules" of the experiment to determine our results. This is also where the ideas of "fairness" and "equally likely" come into play.

*Equally likely* is the idea that there is the same chance of each outcome occurring. If every outcome in an experiment is equally likely, then the experiment is said to be *fair*.

- On which of these spinners does each outcome have an equally likely chance?

The spinner on the top has an *equally likely* chance of spinning each color because each color occupies the same amount of space on the spinner. This paves the way for the main idea of this lesson — the idea of *theoretical probability*.

In theoretical probability, we want to calculate the ratio of something happening to the total number of equally likely outcomes. This is usually written as:

Theoretical Probability | = | number of ways we can have a success or an event happening |

total number of equally likely outcomes |

Let's try this with some examples.

An experiment consists of rolling a six-sided die. Find the theoretical probability of each outcome below. Remember to always reduce your fractions!

- Rolling a 3:

number of ways an event can happen | = | 1 | |

total number of equally likely outcomes | 6 |

- Rolling a number greater than 3:

number of ways an event can happen | = | 3 | = | 1 | |

total number of equally likely outcomes | 6 | 2 |

An experiment consists of flipping a coin 10 times. The table shows the theoretical and experimental probability of getting heads: *P*(heads):

P(heads) | P(tails) | P(heads) + P(tails) | |||||||||||||

Experimental Probability | 3 | 7 | 3 | + | 7 | = | 10 | = | 1 | ||||||

10 | 10 | 10 | 10 | 10 | |||||||||||

Theoretical Probability | 1 | 1 | 9 | + | 1 | = | 2 | = | 1 | ||||||

2 | 2 | 9 | 2 | 2 |

The sum of the probability of heads and the probability of tails is always 1. This is because it is certain that one of the two outcomes will occur.

The *complement* of an event is all the outcomes that are *not* in the event.

The sum of an event and its complement is always 1 or 100% since these are the only possibilities.

P(something happening) + P(something not happening) = 1

We can use this idea in many aspects of probability to solve for unknown variables. For example:

The probability of it snowing is ^{3}⁄_{10} or 30%. We can use algebra to find the probability of it *not* snowing.

P(snow) + P(not snowing) = 1

3 | + | P(not snowing) | = | 10 | |

10 | 10 |

Which we can solve to equal:

P(not snowing) | = | 7 | or | 70% | |

10 |

Another way to describe the probability of an event is to give the *odds* or likelihood of an event.

This is usually written with a colon between the two probabilities, since it is a ratio and is as widely used as the standard form of probability.

However, should you encounter a ratio, pay close attention to the wording of the problem or statement because there are two ways to write odds that could drastically change the probability.

Odds in favor | = | number of ways an event can happen (a) | or | a:b |

number of ways an event can NOT happen (b) |

Odds in favor | = | number of ways an event can NOT happen (b) | or | b:a |

number of ways an event can happen (a) |

Going back to the die:

- What are the odds in favor of rolling a three? 1:5
- What are the odds against rolling a three? 5:1

Turning probability into odds or ratio form is quite simple with a little bit of practice, which we will do in the next section.

We have discussed how to determine and represent the theoretical probability of an experiment's success, and how to convert an experiment's probability into the ratio of odds in favor and against.

Now, let's practice creating the theoretical probability.