Empirical probability is also known as experimental probability. The empirical probability gives the likelihood of an event to happen based on the outcomes from a set of previous experiments. It is different from theoretical probability as empirical probability is dependent on the occurrence of events in a collected data.

Empirical probability of event E, P(E) = $\frac{Number\ of\ occurrences\ of\ E}{Total\ number\ of\ experiments}$

**Example:**

A coin was tossed 4 times and gives a head only once. Find the empirical and theoretical probability of getting a head if the coin is tossed again.

**Solution:**

Theoretical probability = $\frac{1}{2}$

Empirical probability = $\frac{1}{4}$

Empirical probability of event E, P(E) = $\frac{Number\ of\ occurrences\ of\ E}{Total\ number\ of\ experiments}$

A coin was tossed 4 times and gives a head only once. Find the empirical and theoretical probability of getting a head if the coin is tossed again.

Theoretical probability = $\frac{1}{2}$

Empirical probability = $\frac{1}{4}$

Given table shows the numbers obtained in throw of a dice in 5 experiments. Find the probability of getting a 3 empirically.

Experiment | 1 | 2 | 3 | 4 | 5 |

Result | 6 | 2 | 4 | 3 | 3 |

Number of experiments = 5

Number of times 3 came = 2

Empirical Probability = $\frac{2}{5}$

A coin is tossed 50 times in which 32 times it gives a tail. Find the empirical and theoretical probability of getting a head.

Solution:

Total number of experiments = 50

Number of times tail has come = 50 - 32 = 18

Empirical Probability = $\frac{18}{50}$ = $\frac{9}{25}$

Total number of possible events = {H, T} = 2

Number of favorable events = {H} = 1

Theoretical probability = $\frac{1}{2}$

The table below shows the grade point obtained by the students in a certain subject. Find the empirical probability of getting grade below 8.

Grade | 5 |
6 |
7 |
8 | 9 |

Number of students | 34 | 22 | 17 |
14 | 8 |

Total number of students = 34 + 22 + 17 + 14 + 8 = 95

Students getting grade less than 8 = 34 + 22 + 17 =73

Empirical probability = $\frac{73}{95}$

In a buffet, 78 out of 95 guests chose vanilla over chocolate ice cream. Find the empirical probability of a random guest opting for vanilla ice cream.

Solution:

Total number of data obtained = 95

Guests choosing vanilla = 78

Empirical probability = $\frac{78}{95}$