Empirical Probability

Probability describes the chance that an uncertain event will occur. Empirical probability is based upon how likely an event has occurred in the past. It is also called experimental probability. It is based on the relative frequency approach. We can get our results from experience rather than from a theory. We employ the empirical probability generating function in constructing a goodness-of-fit test for negative binomial distributions. In empirical probability, the experimental conditions may not remain the same for all repetitions on that experiment. In statistical terms, the empirical probability is just an estimate of an event.

Formula for Empirical Probability

Empirical Probability = Number of times an event occurred / Total number of trails

Difference Between Empirical Probability and Theoretical Probability

Empirical probability defines a probability value gained from performing an experiment. For example, we want to find out the probability of getting an even number when dice are tossed. To find the probability, we will perform an experiment in which we will toss the dice 100 times and calculate probability from there. 

Empirical Probability = Number of times an event occurred / Total number of trails

Suppose we obtained 60 times an even number during tossing of the dice 100 times, the probability will then be:

P(H) = 60 / 100 = 0.6

Therefore, there is a 0.6 likelihood of obtaining an even number when a dice is tossed 100 times. On the other hand, theoretical probability comes into play when it is not feasible to perform an experiment to determine probability. Then we assume the outcomes of an event are all equally likely. For example, we want to find out whether we obtain an even number when a coin is tossed. When a dice is tossed, there is a 50/50 chance of obtaining an even number or an odd number. Then the probability will be:

P(E) = number of successful outcomes of the event / total number of outcomes

Here, the total number of outcomes is 6, and the number of successful outcomes will be 3(i.e, 2, 4, 6) therefore the probability of occurrence of an even number is:

P(T) = 3 / 6 = 1 / 2 = 0.5

Therefore, there is a 0.5 likelihood of obtaining an even number when a dice is tossed. So finally we can conclude that theoretical probability is based on the assumption that outcomes have an equal chance of occurring while empirical probability is based on the observations of an experiment.

Examples

Example 1. You have conducted a taste test of 100 people that reveals 65 people prefer apple and the remaining prefer banana. Find the empirical probability a person prefers apple over the banana?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 65 / 100 = 0.65

The empirical probability of person preferring apple over banana is 0.65

Example 2. A coin is tossed 5 times and all the three times head showed up. What is the empirical probability of showing a tail when the coin is tossed?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 0 / 5 = 0

The empirical probability of getting a tail is 0.

Example 3. A coin is tossed 2 times and all the three times head showed up. What is the empirical probability of showing a head when the coin is tossed?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 2 / 2 = 1

The empirical probability of getting a head is 1

Example 4. In a dinner for which 120 people attended, 80 people preferred mushrooms and others preferred panners. What is the empirical probability of a person to choose mushroom?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 80 / 120 = 2 / 3 = 0.67

The empirical probability of a person to choose mushroom is 0.67

Example 5. A dice is tossed 10 times and the recordings are recorded in the following table.

Outcome	1	2	3	4	5	6
Frequency	3	2	0	1	3	1

Find the probability of getting a number 4 when the dice is thrown?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 1 / 10 = 0.1

The empirical probability of getting a number 4 when dice is tossed is 0.1

Example 6. There are four marbles in a box and they are of distinct colors red, yellow, green, and blue. One ball is picked each time and this is done 40 times. The observations are recorded in the following table.

Outcome	Red	Yellow	green	blue
Frequency	15	12	6	7

Find the probability of getting a blue ball when a ball is drawn at random?

Solution:

P(H) = Number of times an event occurred / Total number of trails

P(H) = 7 / 40 = 0.175

The empirical probability of getting a number blue ball is 0.175

Advantages of Empirical Probability

• It is free from the hypothesis.
• We need not assume about data.
• Probability is backed by experimental studies and data.
• Covers more cases than classical probability.
• Can be applied when outcomes are not equally likely.

Disadvantages of Empirical Probability

• We need to have large sample sizes.
• Using small sample sizes reduces accuracy
• We may come up with incorrect solutions.
• Repeating the identical experiment an infinite number of times is physically impossible.
• It doesn’t agree with classical probability.