About Experimental Probability
Defining Probability
The mathematics of chance is known as probability (p). The probability of occurrence of an event (E) is revealed by probability.
The probability of an event can be expressed as a number between 0 and 1.
The likelihood of an impossibility is zero. A probability between 0 and 1 can be attributed to any other events that fall in between these two extremes. Experimental probability is the probability that is established based on the outcomes of an experiment. The term ‘empirical probability’ is also used for the same concept.
How Precisely do we Define Experimental Probability?
A probability that has been established by a series of tests is called an experimental probability. To ascertain their possibility, a random experiment is conducted and iterated over a number of times; each iteration is referred to as a trial.
The goal of the experiment is to determine the likelihood of an event occurring or not.
It could involve spinning a spinner, tossing a coin, or using a dice. The probability of an event is defined mathematically as the number of occurrences of the event divided by the total number of trials.
Formulation
The number of times an event occurred during the experiment divided by all the times the experiment was run is known as the experimental probability of that event. Each potential result is unknown, and the collection of all potential results is referred to as the sample space.
Experimental probability is calculated using the following formula:
\(P(E)=\frac{Number~of~times~an~event~occurred~during~an~experiment}{The~total~number~of~times~the~experiment~was~conducte}\)
\(P(E)=\frac{n(E)}{n(S)}\)
Where,
n(E) = Number of events occurred
n(S) = Number of sample space
Solved Experimental Probability Examples
Example 1: The owner of a cake store is curious about the percentage of sales of his new gluten-free cupcake line. He counts the number of cakes that were sold on one day of the week, Monday, where he sold 30 regular and 70 gluten free cakes.Calculate the probability in this case.
Solution:
According to the details in the question, the number of gluten free cakes is n(E) = 70 cakes.
Total number of cakes n(S) = 30 + 70 = 100 cakes.
Substituting these values in the formula.
\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{70}{100}\) = 0.7 = 70%
Hence, the owner of the cake store finds that the gluten-free cupcakes will probably make up 70% of his weekly sales.
Example 2: A baseball manager is interested to know the probability that a prospective new player will hit a home run in the game’s first at-bat. The player has 11 home runs in 1921 games throughout his career. Calculate the probability of the player hitting a home run.
Solution:
The data provided is, the player has hit 11 home runs, n(E) = 11
Total number of games, n(s) = 1921 games.
Substituting these values in the formula.
\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{11}{1921}\) = 0.005726 = 0.5726%
He will therefore have a 0.5726 percent chance of hitting a home run in his first at-bat.
Example 3: A vegetable gardener is checking the likelihood that a fresh bitter gourd seed would germinate. He plants 100 seeds, and 57 of them sprout new plants. Calculate the probability in this scenario.
Solution:
According to the question, the number of bitter gourd plants that sprouted is n(E) = 57.
Total number of seeds sown, n(S) = 100 seeds.
Substituting these values in the formula.
\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{57}{100}\) = 0.57 = 57%
Hence, the probability that a new bitter gourd seed will be sprout is 57%.
Example 4: Joe’s Bagel Shop sold 26 bagels in one day, 9 of which were raisin bagels. Calculate the percentage of raisin bagels that will be sold the following day using experimental probability.
Solution:
As stated in the question, the number of raisin bagels, n(E) = 9.
Total number of bagels Joe sold, n(s) = 26.
Substituting these values in the formula.
\(P(E)=\frac{n(E)}{n(S)}\) = \(\frac{9}{26}\) = 0.346 = 34.6%
As a result, there is a 34.6 percent chance that Joe will sell raisin bagels the following day.
Yes, the ratio obtained is simplified after the ratio between the frequency of the occurrence and the total number of trials is determined.
Compared to experimental probability, theoretical probability is more precise. Only if there are more trials, then the results of experimental probability will be close to the results from theoretical probability.
Actual tests and recordings of events serve as the foundation for calculating the experimental probability of an event. It is determined by dividing the total number of trials by the number of times an event occurred.
A ‘1’ has a 1/6 experimental probability of rolling. Six faces, numbered from 1 to 6, make up a dice. Any number between 1 and 6 can be obtained by rolling the dice, and the likelihood of getting a 1 is equal to the ratio of favorable results to all other potential outcomes, or 1/6.