Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.
To understand this better, imagine flipping a coin. The theoretical probability of landing heads is 50% or 1/2. However, if you actually flip the coin 100 times and record the outcomes, you might get heads 48 times. The experimental probability of getting heads would then be 48/100 or 0.48.
In this article, we will explore the concept of experimental probability, its significance, and how it differs from theoretical probability. We will discuss the formula for calculating experimental probability, provide examples to illustrate its application.
Table of Content
What is Probability?
The branch of mathematics that tells us about the likelihood of the occurrence of any event is the probability. Probability tells us about the chances of happening an event.
The probability of any element that is sure to occur is One(1) whereas the probability of any impossible event is Zero(0). The probability of all the elements ranges between 0 to 1.
There are two ways of studying probability that are
- Experimental Probability
- Theoretical Probability
Now let's learn about both in detail.
What is Experimental Probability?
Experimental probability is a type of probability that is calculated by conducting an actual experiment or by performing a series of trials to observe the occurrence of an event. It is also known as empirical probability.
To calculate experimental probability, you need to conduct an experiment by repeating the event multiple times and observing the outcomes. Then, you can find the probability of the event occurring by dividing the number of times the event occurred by the total number of trials.
Formula for Experimental Probability
The experimental Probability for Event A can be calculated as follows:
P(E) = (Number of times an event occur in an experiment) / (Total number of Trials)
Examples of Experimental Probability
Now, as we learn the formula, let's put this formula in our coin-tossing case. If we tossed a coin 10 times and recorded a head 4 times and a tail 6 times then the Probability of Occurrence of Head on tossing a coin:
P(H) = 4/10
Similarly, the Probability of Occurrence of Tails on tossing a coin:
P(T) = 6/10
What is Theoretical Probability?
Theoretical Probability deals with assumptions in order to avoid unfeasible or expensive repetition experiments. The theoretical Probability for an Event A can be calculated as follows:
P(A) = Number of outcomes favorable to Event A / Number of all possible outcomes
Now, as we learn the formula, let's put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail.
Hence, The Probability of occurrence of Head on tossing a coin is
P(H) = 1/2
Similarly, The Probability of the occurrence of a Tail on tossing a coin is
P(T) = 1/2
Experimental Probability vs Theoretical Probability
There are some key differences between Experimental and Theoretical Probability, some of which are as follows:
Aspect of Difference | Experimental Probability | Theoretical Probability |
|---|---|---|
| Definition | Empirical probability obtained by conducting experiments or observations | Probability obtained by using mathematical principles and formulas |
| Basis | Observed outcomes in real-life experiments | Theoretical predictions based on assumptions and models |
| Accuracy | Can be highly variable due to small sample sizes or other factors | More accurate and reliable, assuming the assumptions and models are correct |
| Calculation | Calculated by dividing the number of times an event occurred by the total number of trials | Calculated by dividing the number of favorable outcomes by the total number of possible outcomes |
| Application | Used when data is collected through experimentation or observation | Used when predicting outcomes for theoretical scenarios |
| Examples | Tossing a coin or rolling a die multiple times to determine the probability of an event | Calculating the probability of drawing a certain card from a deck or the probability of winning a game with specific rules |
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Solved Examples of Experimental Probability
Example 1. Let's take an example of tossing a coin, tossing it 40 times, and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.
Answer:
Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome First
H
Eleventh
T
Twenty-first
T
Thirty-first
T
Second
T
Twelfth
T
Twenty-second
H
Thirty-second
H
Third
T
Thirteenth
H
Twenty-third
T
Thirty-third
T
Fourth
H
Fourteenth
H
Twenty-fourth
H
Thirty-fourth
H
Fifth
H
Fifteenth
H
Twenty-fifth
T
Thirty-fifth
T
Sixth
H
Sixteenth
H
Twenty-sixth
H
Thirty-sixth
T
Seventh
T
Seventeenth
T
Twenty-seventh
T
Thirty-seventh
T
Eighth
H
Eighteenth
T
Twenty-eighth
T
Thirty-eighth
H
Ninth
T
Nineteenth
T
Twenty-ninth
T
Thirty-ninth
T
Tenth
H
Twentieth
T
Thirtieth
H
Fortieth
T
The formula for experimental probability:
P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4
Similarly,
P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6
P(H) + P(T) = 0.6 + 0.4 = 1
Note: Repeat this experiment for 'n' times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get 0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.
Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.
Answer:
Experimental Probability = 30/1000 = 0.03
0.03 = (3/100) × 100 = 3%
The probability that you will buy a defective phone is 3%
⇒ Number of defective phones next month = 3% × 50000
⇒ Number of defective phones next month = 0.03 × 50000
⇒ Number of defective phones next month = 1500
Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?
Answer:
Now the number of people who do not like electric cars is 1000000 - 300000 = 700000
Experimental Probability = 700000/1000000 = 0.7
And, 0.7 = (7/10) × 100 = 70%
The probability that someone chose randomly does not like the electric car is 70%
The probability that someone like electric cars is 300000/1000000 = 0.3
Let x be the number of people who love electric cars
⇒ x = 0.3 × 320 million
⇒ x = 96 million
The number of people who love electric cars is 96 million.
Practice Problems on Experimental Probability
Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?
Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?
Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?
Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?
Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?
Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?