To understand probability, it is important to understand the foundations. In this lesson, you will learn about relative frequency and the foundations of understanding probability.

## Classical Approaches to Probability

Edward is playing Pretzel with his friends. Pretzel is a game that has different colored squares on a mat where each player places a hand or a foot on a different color depending on the spinner. There are two different spinners, one is labeled right hand, left hand, right foot and left foot.

The second spinner is labeled with colors; there are four purple squares, three blue squares, four pink squares and three orange squares randomly arranged on the spinner. Can Edward and his friends predict which spinner will land on which outcome?We don’t have the gift of foresight; we can’t predict the future. So, how do we lowly humans make any sort of educated guess on the outcome of events? We use probability! **Probability** is the likelihood of a certain event occurring out of a total possible number of events. We can use the possible outcomes of a scenario and compare those outcomes to the desired outcome to get an idea of how likely it is that something is going to happen. We can use words like ‘more likely than,’ ‘less likely than’ and ‘equally likely’ to describe the probability of events.For example, what is the probability that the spinner will land on purple? How would you figure this out? First, how many total squares are there? Right, there are 14 total squares: four purple squares, three blue squares, four pink squares and three orange squares.

Next, what is the desired outcome? The desired outcome would be the spinner landing on a purple square. The total possible desired outcome is four because there are four purple squares. Therefore, our ratio would be: 4/14, or approximately 29%.Okay, now find the probability of the spinner landing on an orange square. The ratio for the probability of the spinner landing on an orange square would be 3/14, or approximately 21%. We can say that landing on an orange square is less likely than landing on a purple square.

## Relative Frequency

Another classical approach to probability is **relative frequency**, which is the ratio of the occurrence of a singular event and the total number of outcomes.

This is a tool that is often used after you collect data. You can compare a single part of the data to the total amount of data collected. Edward and his friends play Pretzel and keep track of the colors that the spinner lands on.This is a frequency table:

Color | Frequency |
---|---|

Purple | 7 |

Blue | 3 |

Pink | 5 |

Orange | 5 |

Total | 20 |

Edward and his friends used the color spinner 20 times.

You can see that the spinner landed on a purple square seven times, a blue square three times, a pink square five times and an orange square five times. Now we can find the relative frequency of the data set by dividing each number by the total like this:

Color | Frequency | Relative Frequency |
---|---|---|

Purple | 7 | 7/20 = 35% |

Blue | 3 | 3/20 = 15% |

Pink | 5 | 5/20 = 25% |

Orange | 5 | 5/20 = 25% |

Total | 20 | 20/20 = 100% |

We can use this information to say that the spinner landed on the purple square 35% of the time, the blue square 15% of the time and the orange and pink squares 25% of the time. A relative frequency table allows you to compare the data to the total number of frequencies. The total of the relative frequency table should add up to one, or 100%.

This is because the spinner was used 20 times, and the spinner landed on a color 20 times.You can also compare your predictions to the relative frequency table like this:

Color | Frequency | Relative Frequency | Theoretical Probability |
---|---|---|---|

Purple | 7 | 7/20 = 35% | 29% |

Blue | 3 | 3/20 = 15% | 21% |

Pink | 5 | 5/20 = 25% | 29% |

Orange | 5 | 5/20 = 25% | 21% |

Total | 20 | 20/20 = 100% | 100% |

This table shows the differences between theoretical and actual probability. **Theoretical probability** is the ratio of the desired outcome and the total number of possible outcomes. If you have the probability of a certain outcome without actually doing the experiment first, then you are working with theoretical probability. If you are working with numbers from a data set based on an experiment, then you are working with actual probability. **Actual probability** is the ratio of successful outcomes and the total number of trials.

Let’s say that Edward’s friend, Beatrice, wanted to conduct an experiment to see how many times the spinner would land on a blue square out of ten spins. Beatrice spins the arrow ten times, and it lands on blue four times. In this case, the total number of spins, ten, would be the number of trials, and the number of successful outcomes would be four, which is the number of times the arrow landed on blue.The frequency and relative frequency columns on the above table represent the actual probability. Each frequency ratio shows us the number of successful outcomes for each color divided by the total number of trials. The theoretical probability shows us what should have happened in the trials. Notice that the spinner landed on the purple square six percent more than what it should have in theory, while the spinner landed on the blue and the pink squares less than the theoretical probability.

## Lesson Summary

A classical approach to probability is to use mathematics to predict the unknown. Since we can’t see into the future, we have to use things like theoretical and actual probability to help us predict and understand data. Remember, **probability** is the likelihood of a certain event occurring out of a total possible number of events.

**Theoretical probability** is the ratio of the desired outcome and the total number of possible outcomes, and **actual probability** is the ratio of successful outcomes and the total number of trials.You can use a **relative frequency table**, showing the ratio of the occurrence of a singular event and the total number of outcomes, to compare and analyze theoretical and actual probability. Remember, probability is just an educated guess; it’s not foolproof.

When you are trying to predict a certain outcome, remember that you are not guaranteed that outcome, even when it’s based on high probability.

## Learning Outcomes

Once you’ve finished with this lesson, you should have the ability to:

- Define probability
- Describe what relative frequency is
- Differentiate between theoretical and actual probability
- Explain how to use a relative frequency table to analyze these two types of probabilities