The Concept of Probability
We use the concept of probability almost every day without even realizing it. We take probability to account for even the smallest decisions that we make every day. We can find the probability of an event if we have some background information about the event and its possible outcomes. This type of probability is known as theoretical probability. We can also find the probability of an event by actually performing the experiment and noting down the outcome. This type of probability is known as experimental probability.
List of Formulas
We use two different formulas to calculate theoretical probability and experimental probability. Even though both formulas are logically similar, the numerator and denominator in both cases are slightly different.
1. \(\text{Theoretical Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
2. \(\text{Experimental Probability}=\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}\)
Theoretical Probability vs Experimental Probability
Theoretical probability describes how likely an event is to occur. On the other hand, experimental probability describes how often an event actually occurred in an experiment. For example, we might get either heads or tails when we toss a coin. The theoretical probability is 0.5 in both cases. But we might not get the same result when we perform an experiment. The probability of getting one of the possible outcomes can vary from 0 to 1. That is, it depends on how the experiment goes.
Theoretical Probability Formula
Theoretical probability can be calculated if we know two things: the number of favorable outcomes and the total number of possible outcomes. The favorable outcomes are the outcomes for which we want to find the probability and the total number of possible outcomes can be found by considering all possible outcomes of the experiment.
\(\text{Theoretical Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
Experimental Probability Formula
Experimental probability is calculated by gathering data from trials or random experiments. Here we need to know the number of times the favorable outcome occurs and the total number of trials conducted.
\(\text{Experimental Probability}=\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}\)
Note: The sum of the probabilities of all outcomes in an event is always 1.
Solved Examples
Example 1: Find the theoretical probability of getting an odd number on throwing a dice.
Solution:
The possible outcomes are 1, 2, 3, 4, 5 and 6 and the total number of possible outcomes is 6.
Here 1, 3 and 5 are odd numbers and that means the number of favorable outcomes is 3.
\(\text{Theoretical Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{3}{6}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{1}{2}\)
So, the theoretical probability of getting an odd number on throwing the dice is \(\frac{1}{2}\).
Example 2: Joe got the following result after throwing a dice 6 times:
4, 3, 4, 2, 6, 1
Using this result, find the probability of getting an odd number on throwing a dice.
Solution:
We need to find the probability based on the experiment.
Here, an odd number occurs twice in the experiment.
So, the number of times the event occurs = 2
Total number of trials = 6
\(\text{Experimental Probability}=\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{2}{6}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{1}{3}\)
So, the experimental probability of getting an odd number is \(\frac{1}{3}\).
Example 3: David is an amateur magician. While practicing some magic tricks, he wanted to find the probability of getting any diamond card when he draws a card from a deck. He calculated the probability without performing the experiment.
After calculating the probability, he went ahead and performed the experiment 10 times. The results are as follows: spade, diamond, club, diamond, spade, heart, heart, diamond, club, and club. He calculated the probability again with the information he collected from the experiment.
Find the probability that he calculated initially and compare it with the probability he calculated after performing the experiment.
Solution:
Total number of cards in a deck = 52
Total number of diamond cards in a deck = 13
So, the number of favorable outcomes = 13
Total number of possible outcomes = 52
\(\text{Theoretical Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{13}{52}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{1}{4}\)
David calculated the theoretical probability and got the value as \(\frac{1}{4}\).
Now, David’s observations after performing the experiment:
Spade, diamond, club, diamond, spade, heart, heart, diamond, club, and club
Total number of trials = 10
Number of times he gets diamonds = 3
\(\text{Experimental Probability}=\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}\)
\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{3}{10}\)
After performing the experiment, David found that the probability of getting a diamond card is \(\frac{3}{10}\).
Now we need to compare these two probabilities: \(\frac{3}{10}\) and \(\frac{1}{4}\),
To compare them let us convert them to like fractions:
\(\frac{1}{4}=\frac{1 \times 5}{4 \times 5}=\frac{5}{20}\)
and,
\(\frac{3}{10}=\frac{3 \times 2}{10 \times 2}=\frac{6}{20}\)
So clearly, \(\frac{6}{20}>\frac{5}{20}\) and hence we can say that \(\frac{3}{10}>\frac{1}{4}\)
Therefore, according to David’s observations:
The experimental probability > The theoretical probability.
Probability is a tool used to find the likelihood of an event occurring. The value of probability for an event ranges between 0 and 1. Higher values of probability indicate higher chances of occurrence.
A trial is a random experiment that can be repeated an infinite number of times and a trial has a well-defined set of possible outcomes.
Theoretical probability is conducted by looking at the possible outcomes of an experiment without actually performing the experiment. The experimental probability is calculated by looking at data collected after conducting an experiment a number of times.
Theoretical probability and experimental probability don’t always have the same values. For small repetitions of an experiment, the two probabilities will likely be different. But the value of experimental probability usually gets closer to the value of theoretical probability as we keep repeating the experiment.
There is a third type of probability known as axiomatic probability. In this approach to finding probability, some axioms (rules) are predefined and then the probability is calculated.
Now that you have learned the concept of experimental probability and theoretical probability, go ahead and check out these concepts:
- Probability of compound events formula
- Statistical measures of center: Mean of data formula
- Statistical measures of center: Median formula and mode of data
- Statistical Measures of variation formulas: range, interquartile range, and outliers in a data