104 LearnersLast updated on June 12th, 2025
Theoretical Probability
Theoretical probability is a concept in statistics. It is used to calculate or analyze the likelihood of an event occurring. The calculations are done based on the known possible outcomes and mathematical principles. Here, the assumption is that all outcomes are equally likely. In this article, we’ll be learning about theoretical probability.
What is Theoretical Probability?
The objective of applying theoretical probability is to predict the likely outcome of an event. The prediction is based on mathematical reasoning and known outcomes, and not on actual experiments. In this process, we assume that all outcomes are equally likely. This technique is used when the outcomes are well-defined and predictable.
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How to Find Theoretical Probability?
Make note of the following steps to find theoretical probability.
Step 1: Make sure the experiment that you are working on is well-defined. Determine the sample space by making a list of every experiment result.
Step 2: Determine which results satisfy the requirements of the event considered.
Step 3: Count all possible outcomes in the sample space.
Step 4: Count all favorable outcomes.
Step 5: Apply the formula:
P(E) = Number of Favorable Outcomes / Total Number of Possible
Outcomes
Step 6: Put the probability in percentage or decimal or fraction format.
What is the Difference Between Theoretical, Experimental, and Empirical Probability?
Tips for Theoretical Probability
Here are some tips to master theoretical probability:
- Know Your Basics:
Memorize the key formulas:
Theoretical probability: P(E) = Favorable outcomes/Total Outcomes
Complement Rule: P(Not A) = 1 − P(A)
- Build Sample Spaces Efficiently:
Use tree diagrams for sequences of events, grids or tables for dice and cards, and Venn diagrams for overlapping sets.
- Estimate Before you Calculate:
Quickly estimate what the probability should be. If your answer is above 1 or negative, you know that there is a mistake.
Common Mistakes and How to Avoid Them in Theoretical Probabilities
When working on theoretical probability, students tend to make mistakes. Here, are some common mistakes and their solutions:
Mistake 1
Confusion Between Theoretical and Experimental Probability
Theoretical probability is what is expected to happen in an ideal situation. We can find it by dividing the total number of desired outcomes by the number of possible outcomes. Experimental probability, as the name suggests, is based on actual results. It is found by dividing the number of times an even happens successfully by the total number of tries (including failed attempts).
Mistake 2
Not Defining Sample Space Correctly
Students must define the sample space correctly. They can use tree diagrams for sequential events, tables or grids for combined events, and Venn diagrams for overlapping events.
Mistake 3
Overcounting or Undercounting Outcomes
Students must use systematic listing methods and double-check their permutations and combinations.
Mistake 4
Ignoring Independence and Dependence of Events
Students must identify the types of event. If the occurrence of an event doesn’t change the chances of another event, they are called independent events. Dependent events are happenings that affect the likely outcomes of two or more events
Mistake 5
Rounding Errors
Students must keep at least 4 decimal places during the intermediate steps before rounding off the final answer.
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Real Life Applications of Theoretical Probability
Theoretical probability can be used in many ways. Here are some examples:
- Gambling and Games of Chance:
We use theoretical probability for predicting outcomes in games like roulette, dice, or cards. It can also be used to calculate the odds of winning.
- Insurance and Risk Assessment:
We use theoretical probability in insurance and risk assessment to forecast stock price movements. We also use it to assess the likelihood of returns or losses on investments.
- Traffic Management and Safety:
We use theoretical probability in traffic management to estimate the probability of collisions at intersections. We also use it to manage traffic signals and reduce congestion. It is also applied by self-driven vehicles to make decisions on the road.
Solved Examples on Theoretical Probability
Problem 1
What is the probability of getting heads when tossing a fair coin?
The probability of getting heads is ½
Explanation
Identify the sample space:
S = {heads, tails}
Total outcomes = 2
Determine favorable outcomes:
Favorable outcome (heads) = 1
Calculate the probability:
P(heads) = 1/2
Problem 2
What is the probability of rolling a 4 on a fair six-sided die?
The probability of rolling a 4 is ⅙ on a fair six-sided die.
Explanation
Identify the sample space:
S = {1, 2, 3, 4, 5, 6}
Total outcomes = 6
Favorable outcome:
Only one outcome is 4.
Probability:
P(4) = 1/6
Problem 3
What is the probability of drawing an Ace from a standard 52-card deck?
The probability of drawing an Ace is 1/13
Explanation
Total outcomes:
There are 52 cards
Favorable outcomes:
Number of aces = 4
Probability:
P(Ace) = 4/52 = 1/13
Problem 4
What is the probability of obtaining a sum of 7 when rolling two fair six-sided dice?
The probability of obtaining a sum of 7 is 1/6
Explanation
Total outcomes:
6 x 6 = 36
Favorable outcomes:
The pairs that sum to 7 are:
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) - 6 outcomes
Probability:
P(sum of 7) = 6/36 = 1/6
Problem 5
A spinner is divided into 8 equal sectors numbered 1 through 8. What is the probability of landing on sector 5?
The probability of landing on sector 5 is 1/8
Explanation
Total outcomes:
8 sectors
Favorable outcomes:
Only sector 5 qualifies
Probability:
P(5) = 1/8
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FAQs on Theoretical Probability
1.What is theoretical probability?
2.How is theoretical probability calculated?
3.What is the range of theoretical probability values?
4.How does theoretical probability relate to probability distributions?
5.How can theoretical probability be applied in real life?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
