Empirical Probability: Probability describes the chance that an uncertain event will occur. Empirical probability is based on 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 of that experiment. In statistical terms, the empirical probability is just an estimate of an event.
Empirical Probability Meaning
Empirical probability refers to the probability of an event based on observed data or experimental results. Unlike theoretical probability, which is calculated using mathematical principles and assumptions about the underlying probability distribution, empirical probability is derived from actual outcomes obtained through observation or experimentation.
Empirical Probability Formula
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 experimenting. 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 the 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 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.
Empirical Probability 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.
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Summary - Empirical Probability
Empirical probability, also known as experimental probability, is all about real-world data and observations. It's like looking at past experiences to figure out how likely something is to happen again. We use it when we can't rely on theoretical calculations and need to see what actually happens in practice. For example, if we want to know the chance of flipping a coin and getting heads, we might flip it many times and see how often heads comes up. This gives us a rough idea of the probability based on what we've seen happen. Empirical probability is flexible because it can handle situations where outcomes aren't equally likely, and it doesn't rely on any fancy theories. But it does have its limitations; we need a lot of data to get accurate results, and sometimes our observations might not match up with what we expect from theoretical probability. Overall, empirical probability is like learning from experience rather than relying on theories, making it a valuable tool in understanding the likelihood of events in the real world.
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