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In probability theory, empirical probability refers to the ratio of the number of outcomes in which an event occurs to the total number of outcomes. It calculates probabilities based on observation and experience.
- Empirical probability is also called relative frequency or experimental probability.
- One of the advantages of experimental probability is that the procedure is free of assumptions.
- Empirical probability depends upon the likelihood of an event that has happened in the past.
- Suppose we flip a coin 50 times and heads show up 25 times, then the empirical probability of getting heads on any flip is 0.5.
- The results of the probability can be improved by adopting assumptions in the form of a statistical model.
- Mathematically, it can be represented as:
Empirical Probability Formula = f/n
where,
- f : number of times an event occurs
- n : total number of trials
Key Terms: Probability, Empirical Probability, Empirical Probability Formula, Event, Outcomes, Relative Frequency, Experimental Probability
What is Empirical Probability?
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Probability refers to the number of times an event can occur. Empirical probability is a type of probability that is based on historical data. It is the ratio of the number of outcomes in which a specific event occurs or occurs to the number of real trials made.
- Empirical probability is not calculated theoretically but based on actual experimental data.
- In a nutshell, experimental probability is a forecast based on real experimental observation.
- The probability is based on the results of an actual experiment conducted many times.
- If we increase the number of trials in an experiment, the theoretical and experimental probabilities will be nearly the same.
- It is the form of an estimator or estimate of a probability.
Empirical Probability ExampleExample: Consider a dice. Dice are cube-shaped objects with six faces and numerals 1 to 6 printed on each face. This implies that just one face will appear each time the dice are rolled.
Experimental Probability |
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Empirical Probability Formula
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Empirical probability of a certain event occurring may be expressed or defined as the estimated likelihood of that specific event occurring in a complete series of occurrences. Mathematically it can be represented as:
Empirical Probability Formula = Number of times an event occured / Total number of times an experiment occured
Empirical Probability Formula = f/n
where,
- f : number of times an event occurs
- n : total number of trials
Example of Empirical Probability FormulaExample: In a group of 100 people, 30 people chose to order non-veg curry over the veg curry. What is the empirical probability of someone ordering veg curry? Ans: Given that: Total number of people = 100 Hence, Therefore, the empirical probability of someone ordering veg curry is 0.70 or 70 %. |
Experimental Probability Formula
Empirical Probability Advantages
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The advantages of empirical probability are as follows:
- Empirical probability is not based on any assumptions or hypotheses.
- The result of the probability is supported by experimental studies and data.
- It gives more number of cases than classical probability.
- Empirical probability can be used when outcomes are not equally likely.
- Probability is backed by experimental studies and data.
Empirical Probability Disadvantages
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The downside of using empirical probabilities is that it produces findings pointing to estimated probabilities that are either extremely near to the figure zero (0) or very close to the figure one (1).
- It should be emphasised that very large sample sizes would be necessary to foresee or anticipate such probability reasonably.
- In general, it has greater accuracy than empirical probabilities if the given data is accurate.
- Empirical probability might provide incorrect solutions.
- It repeats an experiment an infinite number of times, which is physically impossible.
Example of Empirical Probability DisadvantagesExample: Consider a case in which the minimum value of the daily maximum rainfall received in a certain location during the summer month of May is less than 20mm.
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Things to Remember
- Empirical Probability is the product of the number of times an event occurs multiplied by the total number of trials.
- This type of probability is backed by experimental studies and does not require any assumptions.
- Empirical probability is used to determine the goodness-of-fit test for negative binomial distributions.
- The result is based on the proportion of tasks attempted to successful completion of the task.
- An event is said to occur when the probability of an event is equal to one.
Sample Questions
Ques: If a fair die was rolled 120 times, determine how many times the number 5 appeared? (3 marks)
Ans: As we all know, each number has an equal chance of occurring, i.e. 1/6.
The likelihood of an event occurring is
120*1/6=20
As a result, the frequency of 5 is 20 out of 120. (on an average).
The experimental or empirical probability of an event is based on what has actually occurred, whereas the theoretical probability of the event seeks to forecast what will occur based on the total number of potential possibilities. We can anticipate the practical and theoretical probability to be approximately equal as the number of trials in an experiment increases.
Ques: A survey of 880 students was performed at a university. 500 of these people identified as "smokers." Calculate the empirical likelihood that a randomly chosen student is not "smoking”? (3 marks)
Ans: Total number of students = 880
Number of students who are smokers = 500
Required probability = \(\frac{880 - 500}{880} = \frac{380}{880} = 0.432 (approx.)\)
Ques: 12 salt packets, each labelled 2 kg, actually contained the following salt weights (in kg):
1.95, 2.02, 2.06, 1.98, 2.03, 1.97, 2.04, 1.99, 1.985, 2.025, 2.00, and 1.98
One package is chosen at random from among these. What is the likelihood that the selected packet contains more than 2 kg of salt? (3 marks)
Ans: Total number of packets = 12
Number of packets containing more than 2 kg of salt = 5
P(E) = \(\frac{Number\,of\,packets\,containing\,more\,than\,2\,kg\,of\,salt}{Total\,number\,of\,packets}\)
= \(\frac{5}{12}\)
Ques: A teacher discovered that 26 people took two to three hours, 14 people took three to four hours, and 10 people spent more than four hours to finish a specific assignment while working on a math project. What is the experimental chance of completing the task in less than four hours? (3 marks)
Ans: Total number of students = 26 +14 + 10 = 50
Number of people who take less than 4 hours to do the project = 26 + 14 = 40
Required probability = \(=\frac{40}{50}\,= \frac{4}{5}\)
Ques: Team A has won 29 of the previous 50 cricket matches between teams A and B. What is the experimental chance that team A will lose the next cricket match between the two teams? (3 marks)
Ans: Total number of matches = 50
Number of matches which team A won = 29
Number of matches which team A lost = 50 - 29 = 21
\(\therefore\) Required probability = \(\frac{Number\,of\,matches\,which\,team\,A\,lost}{Number\,of\,matches\,which\,team\,A\,won} = \frac{21}{50}\)
Ques: A teacher discovered that 20 people took two to three hours, 10 people took three to four hours, and 30 people spent more than four hours to finish a specific assignment while working on a math project. What is the experimental chance of completing the task in less than four hours? (3 marks)
Ans: Total number of students = 20 +10 + 30 = 60
Number of people who take less than 4 hours to do the project = 20 + 10 = 30
Required probability = \(=\frac{30}{60}\,= \frac{1}{2}\)
Ques: Team A has won 30 of the previous 100 cricket matches between teams A and B. What is the experimental chance that team A will lose the next cricket match between the two teams? (3 marks)
Ans: Total number of matches = 100
Number of matches which team A won = 30
Number of matches which team A lost = 100 - 30 = 70
\(\therefore\) Required probability = \(\frac{Number\,of\,matches\,which\,team\,A\,lost}{Number\,of\,matches\,which\,team\,A\,won} = \frac{70}{100}\)
Ques: A coin toss four times and the result was four heads. Using the empirical probability formula find out what is the empirical probability of getting a head? (3 marks)
Ans: Given that
Total number of trials = 4
Number of heads = 4
Hence, Empirical probability = 4 / 4 = 1.
Therefore, the empirical probability of getting a head is given by 1 or 100%.
Ques: In a buffet, 40 out of 200 people chose to order coffee over soup. What is the empirical probability of someone ordering soup? (2 marks)
Ans: Given that
Total number of people = 200
Number of people who choose coffee or tea = 40.
Number of people who choosing coffee or tea = 200 - 40 = 160
Hence, empirical probability is given as 160 / 200 = 0.8
Ques: A survey of 800 students was performed at a university. 600 of these people identified as "smokers." Calculate the empirical likelihood that a randomly chosen student is not "smoking”? (3 marks)
Ans: Total number of students = 800
Number of students who are smokers = 600
Required probability = \(\frac{800 - 600}{800} = \frac{200}{800} = 0.25 (approx.)\)
Ques: In a buffet, 140 out of 150 people chose to order coffee over soup. What is the empirical probability of someone ordering soup? (2 marks)
Ans: Given that
Total number of people = 150
Number of people who choose coffee or tea = 140.
Number of people who choosing coffee or tea = 150 - 140 = 100
Hence, empirical probability is given as 10 / 150 = 0.067
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