### Introduction to Empirical Probability

An empirical probability (also called experimental probability) is calculated by collecting data from past trials of the experiments. The experimental probability obtained is used to predict the future likelihood of the event occurring.

### Formula and Examples of Empirical Probability

To calculate an empirical/ experimetnal probability, one can use the formula

$$P(A)=\frac{\text{Number of trials in which $A$ occurs} }{$\text{Total number of trials}}$$

**Coin Flip**: Let us flip a coin 200 times and get heads 105 times. The empirical probability of getting heads is $\frac{105}{200} = 0.525%, or 52.5%.**Weather Prediction:**Let you track the weather for a month and see that it rained 12 out of 30 days. The empirical probability of rain on a given day that month is $\frac{12}{30} = 0.4$ or 40%.**Plant Growth:**Let you plant 50 seeds and 35 sprout into seedlings. The experimental probability of a seed sprouting is $\frac{35}{50} = 0.70$ or 70%.**Board Game:**Suppose you play a new board game 10 times and win 6 times. The empirical probability of winning the game is $\frac{6}{10} = 0.6$ or 60%.**Customer Preferences:**In a survey of 100 customers, 80 prefer chocolate chip cookies over oatmeal raisins. The empirical probability of a customer preferring chocolate chip cookies is $\frac{80}{100} = 0.80$ or 80%.**Basketball Game:**A basketball player practices free throws and makes 18 out of 25 attempts. The experimental probability of the player making their next free throw is $\frac{18}{25} = 0.72$ or 72%.

### Empirical Probability From Frequency Tables

From a frequency table, one can calculate the probability that a certain data value falls into any data group/ class. For example, consider the frequency table of examination scores in a certain class.

Class | Frequency ($f$) | $frf$ |
---|---|---|

40 – 49 | 1 | $\frac{1}{20}=0.05$ |

50 – 59 | 2 | $\frac{1}{20}=0.10$ |

60 – 69 | 3 | $\frac{3}{20}=0.15$ |

70 – 79 | 4 | $\frac{4}{20}=0.20$ |

80 – 89 | 6 | $\frac{6}{20}=0.30$ |

90 – 99 | 4 | $\frac{4}{20}=0.20$ |

Let event $A$ is the event that a student scores between 90 and 99 on the exam, then

$$P(A) = \frac{\text{Number of students scoring 90-99}}{\text{Total number of students}} = \frac{4}{20} = 0.20$$

Notice that $P(A)$ is the relative frequency of the class 90-99.

### Key Points about Empirical Probability

- It is based on actual data, not theoretical models.
- It is a good approach when the data is from similar events in the past.
- The more data you have, the more accurate the estimate will be.
- It is not always perfect, as past results do not guarantee future outcomes.

### Empirical Probability also has Limitations

- It can be time-consuming and expensive to collect enough data.
- It may not be representative of the future, especially if the underlying conditions change.