About Theoretical Probability
Theoretical Probability
The term ‘theoretical probability’ refers to the theory related to probability. Based on math and logic, it predicts the outcome of the occurrence of an event. It informs us of what should happen in an ideal scenario without performing any experiments. Probability is used to control traffic flow through a highway system, a phone exchange, or a computer processor.
Theoretical probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. It is not necessary to conduct an experiment to determine the theoretical probability. However, situational knowledge is required to estimate the probability of an event happening. Theoretical probability assumes that all events are equally likely to occur and makes predictions about the likelihood that an event will take place.
Example: Consider the scenario where we need to determine the likelihood of drawing 6 cards when there are a total of 12 cards. The probability is then calculated as 0.5 by dividing the number of favorable events, which is 6, by the total number of possible outcomes, which is 12.
Thus, the theoretical probability formula is expressed as follows:
Theoretical probability = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\)
Example: Suppose 120 lottery tickets were sold overall, and Joseph holds 15 of them. Find the probability of Joseph winning the lottery.
Here, the number of favorable outcomes = 15
Number of possible outcomes = 120
Theoretical probability = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}=\frac{15}{120}\) = 0.125
Experimental Probability
The probability based on the repeated trials of an experiment is called experimental probability.
Experimental probability = \(\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}\)
Example: After spinning a penny 15 times, we get 6 heads and 9 tails. Find the probability of spinning heads.
Total number of trials = 15
Number of times heads appeared = 6
Experimental probability = \(\frac{\text{Number of times the event occurs}}{\text{Total number of trails}}=\frac{6}{15}\) = 0.4
Solved Theoretical Probability Examples
Example 1: Rachel threw a dice hoping that she would get the number 4. What is the probability that Rachel gets the number 4?
Solution:
The possible outcomes of rolling a dice are 1, 2, 3, 4, 5, and 6.
So, the total number of outcomes = 6
Rachel wants the number 4.
So, the number of favorable outcomes = 1
Theoretical probability = \(\frac{\text{Number of favorable outcomes }}{\text{Number of possible outcomes}}\) [Formula]
Probability (3) = \(\frac{1}{6}\) = 0.167 [Substitute the values]
Therefore, the probability of Rachel getting a 4 is 0.167.
Example 2: The letters of the word ‘THEORETICAL’ are put in a bag. How likely is it that the letter “E” will be taken out of the bag when picked randomly?
Solution:
The total number of letters = 11
Since the bag has two Es, the number of favorable outcomes = 2.
Theoretical probability = \(\frac{\text{Number of favorable outcomes }}{\text{Number of possible outcomes}}\) [Formula]
Probability = \(\frac{2}{11}\) = 0.182 [Substitute the values]
Hence, the probability of drawing the letter ‘E’ is 0.182.
Example 3: Calculate the probability of selecting a pink ball if a bag includes 9 pink balls and 18 yellow balls.
Solution:
The formula given below is used to compute the theoretical probability.
Theoretical probability = \(\frac{\text{Number of favorable outcomes }}{\text{Number of possible outcomes}}\)
Number of favorable outcomes = 9
Number of possible outcomes = 9 + 18 = 27
Probability (Pink) = \(\frac{9}{27}\) = 0.33
Therefore, the probability of picking up a pink ball is 0.33.
To calculate the theoretical probability, perform the following steps:
- Count the number of favorable or desirable outcomes.
- Count all possible outcomes.
- Divide the outcome of step 1 by the outcome of step 2.
The probability calculated based on the repeated trials of an experiment is called experimental probability.
When it is not feasible to conduct an experiment, the theoretical probability is computed. It provides a reasonable indication of the possibility of an outcome. On the other hand, the experimental probability is computed using data from repeated experiments.