There are several circumstances in which we would predict the outcome of an event in real life. We may be certain or uncertain about the outcome of an event. The certainty of an event lies between $0$ and $1$, or equivalently between $0 \%$ and $100 \%$, with $0$ representing an impossibility and $1$ representing absolute certainty. In such instances, we believe that there is a probability that this event will happen. Probability is the branch of mathematics, which refers to the occurrence of a random experiment.

There are four main types of approaches to computing probability. These are Theoretical, Experimental, Subjective, or Axiomatic approaches. Let’s understand what is experimental probability with examples.

## What is Experimental Probability?

Experimental Probability, also known as Empirical Probability is a type of probability that is based primarily on a set of experiments. A random experiment is repeated many times to determine their likelihood; each repetition is known as a trial. The experiment is conducted to find the chance of an event occurring or not occurring. The random experiment can be tossing a coin, rolling a die, or rotating a spinner.

Mathematically, the experimental probability of an event is equal to the number of times an event occurred divided by the total number of trials. For instance, if you toss a coin $50$ times and record the result i.e., a head or a tail. Then the experimental probability of obtaining a head is calculated as a fraction of the number of times the head appeared and the total number of tosses.

## Experimental Probability Formula

The experimental probability of an event occurring is calculated by dividing the number of times the event occurred during the experiment by the total number of times the experiment was conducted.

As a result, each possible outcome is uncertain, and the sample space is the collection of all possible outcomes. The Experimental Probability Formula assists us in calculating the experimental probability, which is calculated as $\text{P}(\text{E}) = \frac{\text{Number of times even occurs}}{\text{Total number of times experiment is performed}}$.

## Experimental Probability Examples

Here are a few experimental probability examples to make you understand the concept better.

**Example 1:** There were dogs of many different breeds at the dog park last Saturday. Use experimental probability to determine that a dog picked at random in the neighborhood will be a German Shepherd.

Dog Breed | Number of Dogs |

Yorkshire Terriers | 3 |

Springer Spaniels | 3 |

Dachshunds | 2 |

German Shepherds | 6 |

Let $\text{E}$ be the event of picking a dog at random.

The total number of dogs $= 3 + 3 + 2 + 6 = 14$

Number of dogs of German Shepherd breed $= 6$

Therefore, the probability of picking a dog that is a German Shepherd is $\text{P}(\text{E}) = \frac{6}{14} = \frac{3}{7}$.

**Example 2:** Calculate the probability that while ordering an exotica pizza, the next order will include a Schezwan Sauce topping. The following can be found on an exotica pizza:

Pizza Toppings | Number of Orders Made |

Pepperoni | 8 |

Cheese | 5 |

Mushrooms | 10 |

Schezwan Sauce | 16 |

Black Olives | 4 |

Let $\text{E}$ be the event of topping preferred in the next order is Schezwan Sauce.

The total number of orders made $= 8 + 5 + 10 + 16 + 4 = 43$

Number of orders with Schezwan Sauce topping $= 16$

Therefore, the probability of pizza with Schezwan Sauce topping $\text{P}(\text{E}) = \frac{16}{43} = 0.3721$.

**Example 3:** The following table shows the observations made after throwing a 6-sided die 80 times.

Outcome | Frequency |

1 | 13 |

2 | 10 |

3 | 15 |

4 | 14 |

5 | 12 |

6 | 16 |

Find the probability of an experiment in a throw of dice of

a) obtaining a 4

b) Obtaining a number less than 4

c) Rolling a 3 or 6

The total number of throws of a die $= 13 + 10 + 15 + 14 + 12 + 16 = 80$

a) Let $\text{E}_1$ be the event of getting 4

Number of event $\text{E}_1$ occurred (4 is obtained) = $14$

Therefore, $\text{P}(\text{E}_1) = \frac{14}{80} = \frac{7}{40} = 0.175$.

b) Let $\text{E}_2$ be the event of getting a number less than 4

Number of event $\text{E}_2$ occurred (number less than 4 is obtained) = $13 + 10 + 15 = 38$

Therefore, $\text{P}(\text{E}_2) = \frac{38}{80} = \frac{19}{40} = 0.475$.

c) Let $\text{E}_3$ be the event of getting a number 3 or 6

Number of event $\text{E}_3$ occurred (number 3 or 6 is obtained) = $15 + 16 = 31$

Therefore, $\text{P}(\text{E}_3) = \frac{31}{80} = 0.3875$.

**Example 3:** The following bar graph summarizes the weather conditions in a city for each day this month so far.

Based on this data, what is a reasonable estimate of the probability that it is sunny tomorrow?

Let $\text{E}$ be the event of sunny tomorrow.

Number of days of observations = $3 + 5 + 4 + 3 = 15$

Number of days it was sunny = $3$

Therefore, $\text{P}(\text{E}) = \frac{3}{15} = \frac{1}{5} = 0.2$.

**Example 4:** A spinner marked with four sections blue, green, yellow, and red was spun 100 times. The results are shown in the table.

Section | Frequency |

Blue | 14 |

Green | 10 |

Yellow | 8 |

Red | 68 |

a) Find the experimental probability of landing on the green.

b) Find the experimental probability of landing on blue or yellow.

c) Find the experimental probability of landing on a colour that is not red.

Total number of times the spinner was spun = $100$.

a) Let $\text{E}_1$ be the event of spinner landing on green.

Number of times the spinner was landing on green = $10$

Therefore, probability of spinner landing on green = $\text{P}(\text{E}_1) = \frac{10}{80} = \frac{1}{8} = 0.125$

b) Let $\text{E}_2$ be the event of spinner landing on green.

Number of times the spinner was landing on blue or yellow = $14 + 8 = 22$

Therefore, probability of spinner landing on blue or yellow = $\text{P}(\text{E}_2) = \frac{22}{80} = \frac{11}{40} = 0.275$.

c) Let $\text{E}_3$ be the event of spinner not landing on red. (Spinner is landing on blue, green, or yellow).

Number of times the spinner was landing on blue, green, or yellow = $14 + 10 + 8 = 32$

Therefore, probability of spinner not landing on red = $\text{P}(\text{E}_3) = \frac{32}{80} = \frac{2}{5} = 0.4$.

**Difference Between Experimental Probability and Theoretical Probability**

The following are the main differences between experimental probability and theoretical probability.

Experimental Probability | Theoretical Probability |

Empirical probability is based on the data which is obtained after an experiment is carried out | Theoretical probability is based on what is expected to happen in an experiment, without actually conducting it |

It is the result of $\frac{\text{Number of occurrences of an event}}{\text{Total number of trials}}$ | It is the result of $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ |

Example: A coin is tossed $10$ times. It is recorded that the coin landed on the head $7$ times and on the tail $3$ times, then $\text{P} ( \text{Head})= \frac{7}{10} = 0.7$ and $\text{P} ( \text{Tail}) = \frac{3}{10} = 0.3$ | Example: A coin is tossed. Sample Space $\text{S } = \{\text{H}, \text{T} \}$ and $\text{P}( \text{Head}) = \frac{1}{2} = 0.5$ and $\text{P}( \text{Tail}) = \frac{1}{2} = 0.5$ |

## Practice Problems

- Suppose you toss a coin 100 times and get a head 58 times. Now, you toss a coin at random. What is the probability of getting a head?
- A coin is tossed 150 times and the head is obtained 71 times. Now, if a coin is tossed at random, what is the probability of getting a tail?
- A bag contains 10 red marbles, 8 blue marbles, and 2 yellow marbles. Find the experimental probability of getting a blue marble.
- A spinner is divided into eight equal sectors, numbered 1 through 8.
- What is the probability of spinning a 2?
- What is the probability of spinning a number from 1 to 4?
- What is the probability of spinning a number divisible by 2?

- Ravi leaves for work at the same time each day. Over a period of 227 working days, on his way to work, he had to wait for a train at the railway crossing on 58 days. Calculate the experimental probability that Ravi has to wait for a train on his way to work.
- Each time Vinay shuffled a pack of cards before a game, he recorded the suit of the top card of the pack. His results for 140 games were 34 Hearts, 36 Diamonds, 38 Spades, and 32 Clubs.
- Find the experimental probability that the top card of a shuffled pack is
- a Heart
- a Club or Diamond

## FAQs

### What is the experimental probability?

Experimental Probability, also known as Empirical Probability is a type of probability that is based primarily on a set of experiments. A random experiment is repeated many times to determine their likelihood; each repetition is known as a trial. The experiment is conducted to find the chance of an event occurring or not occurring. The random experiment can be tossing a coin, rolling a die, or rotating a spinner.

### What is the formula for experimental probability?

The formula for experimental probability for an event $\text{E}$ is given by $\text{P}(\text{E}) = \frac{\text{Number of occurrences of an event}}{\text{Total number of trials}}$.

### Why is it called the experimental probability?

It is called an experimental probability because a random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event occurring or not occurring.

### What are experimental and theoretical probabilities?

The two most commonly used approaches of find the probability of an event are

a) **Empirical probability** is based on the data which is obtained after an experiment is carried out.

b) **Theoretical probability** is based on what is expected to happen in an experiment, without actually conducting it

### Can we use experimental probability in real life?

Experimental probability is widely used in real life. For example, the meteorological department uses the concept of experimental probability to forecast weather conditions based on past records.

## Conclusion

Experimental Probability, also known as Empirical Probability is a type of probability that is based primarily on a set of experiments. A random experiment is repeated many times to determine their likelihood; each repetition is known as a trial. The experiment is conducted to find the chance of an event occurring or not occurring. The formula used to calculate the experimental probability of an event $\text{E}$ is given by$\text{P}(\text{E}) = \frac{\text{Number of times even occurs}}{\text{Total number of times experiment is performed}}$.