What is Theoretical Probability Definition Formula and How to Calculate with Examples
Probability in literal terms means the chances of occurrence of an event i.e the possibility of happening of an event. Academically, you will learn probability as a branch of mathematics that deals with the occurrence of a random event.
We cannot predict many events with total certainty. As we can predict only the chance of an event to occur i.e. how likely they are to happen. Also, Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. For example: what is the likelihood of a two-sided one-rupee coin when tossed in the air? There are two possible outcomes, head, and tail.
In this particular article, we shall be discussing in detail the following concepts -
Introduction
Theoretical Probability - Definition and example
Experimental Probability with example
Theoretical Probability vs Experimental Probability
Key learnings
Frequently asked questions
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Theoretical Probability Definition
The theoretical probability math definition states that it is related to the theory behind probability. In theoretical probability, we utilize the knowledge of a situation to calculate the probability of an event. We do not conduct any experiment; instead, we just use the knowledge of a situation. The theoretical probability formula is as follows: it states that the probability of occurrence of an event is equal to the number of favorable outcomes divided by the total number of outcomes that are possible.
The mathematical formula of how we define theoretical probability is:
P(E)=The count of favorable outcomes/Total number of possible outcomes
The count of favorable outcomesTotal number of possible outcomes.
Theoretical Probability Examples
Let us have a look at some theoretical probability questions:
1. Find the probability that when a fair die is rolled, it rolls a 4.
Answer: Here, the total number of possible outcomes is 6.
Number of favourable outcomes = number of times a fair die can roll to a 4 in a single throw = 1
According to the formula of theoretical probability, ‘
P(E)=The count of favorable outcomes/Total number of possible outcomes
The count of favorable outcomesTotal number of possible outcomes.
So, P (a fair die rolls a 4 in a throw) = 1/6
2. A fair die is rolled. Find out the probability that the die rolls up to an odd number.
Answer: Here, the total number of possible outcomes is 6.
Number of favourable outcomes = number of times a fair die can roll to an odd number in a single throw.
Total outcomes of a fair die = {1,2,3,4,5,6}
Favourable outcomes (Odd numbers) = {1,3,5} =3
So, a number of favorable outcomes = 3.
According to the formula of theoretical probability, ‘
P(E)= The count of favorable outcomes/Total number of possible outcomes.
The count of favorable outcomesTotal number of possible outcomes.
So, P (die rolls up to an odd number) = 3/6 = 1/2 .
What is Experimental Probability?
It is also known as empirical probability. It is calculated on the basis of the performance of actual experiments or trials and their outcomes. Experiments are conducted in a serial manner. These are called random experiments as the results of these experiments are unpredictable. The experiments are carried out a number of times to determine the outcomes.
The mathematical formula of how we define experimental probability is:
P(E) =the number of times event E occurs/ total number of trials of the experiment.
Solved Problems
Let us have a look at some experimental probability questions: -
1. Two friends A and B toss a fair coin 10 times in a row. The outcomes for this experiment are as follows:
Find the experimental probability for each outcome.
Answer: According to the formula of experimental probability,
P(E) =the number of times event E occurs/total number of trials of the experiment
the number of times event E occurs a total number of trials of the experiment.
Now, P (Occurrence of heads) =number of times head occurs/total number of trials.
P (Occurrence of tails) = a number of times tails occur/ total number of trials.
Calculation of Experimental Probability
Theoretical Probability vs Experimental Probability
When comparing experimental and theoretical probability, we should clearly look at their definitions to understand the fundamental difference between the two. In the case of experimental probability, we perform experiments repeatedly to get to know the outcomes and calculate the probability of those series of outcomes. These experiments are known as random experiments as the results of these experiments are unpredictable. The collection of outcomes is what constitutes an event. If the outcomes have equal chances of happening, the event is termed as an equally likely event. Each repetition for conducting the experiment is called a trial. By the definition of probability, we can write this formula for the calculation of the probability of an event:
P(E)=The count of favorable outcomesTotal number of possible outcomes
The count of favorable outcomesTotal number of possible outcomes.
When in the case of experimental probability, the number of trials is extremely high, the experimental probability then starts approaching the theoretical probability values. The theoretical probability meaning is when the probability is calculated by utilising the knowledge of a certain situation and not carrying out the experiment actually.
In real life, there are some situations when carrying out experiments is not feasible, or it is too expensive to carry out those experiments. In such cases, theoretical probabilities are calculated to have a fair idea of how likely the outcomes are to occur and so that necessary steps or precautions can be taken to avoid dangerous situations. For example, when we launch a satellite, the probabilities calculated are theoretical and not experimental.
Key Learnings From the Chapter
Probability is of two types are theoretical probability and experimental probability
In theoretical probability theory is used to find out the probability
To find the chance of occurrence of an event the actual experiments or trials are considered in an experimental probability
When the number of events that need to be compared is larger, the theoretical probability is used
To get more precise and reliable results theoretical probability is used
Both types have their own advantages and disadvantages.
FAQs on Theoretical Probability Complete Guide with Formula and Examples
1. What is theoretical probability?
The theoretical probability of an event is calculated using the formula Favourable Outcomes ÷ Total Possible Outcomes when all outcomes are equally likely.
- Formula: P(E) = Number of favourable outcomes / Total number of possible outcomes
- It is based on reasoning rather than experiment.
- Example: For a fair die, probability of rolling 3 = 1/6.
2. What is the formula for theoretical probability?
The formula for theoretical probability is P(E) = n(E) / n(S), where n(E) is favourable outcomes and n(S) is total outcomes.
- P(E) = probability of event E
- n(E) = number of favourable outcomes
- n(S) = total outcomes in the sample space
3. How do you calculate theoretical probability step by step?
To calculate theoretical probability, divide the number of favourable outcomes by the total number of possible outcomes.
- Step 1: Identify the sample space (S).
- Step 2: Count favourable outcomes n(E).
- Step 3: Apply P(E) = n(E)/n(S).
4. What is the difference between theoretical and experimental probability?
The key difference is that theoretical probability is calculated using a formula, while experimental probability is based on actual results from experiments.
- Theoretical: Favourable outcomes ÷ Total outcomes
- Experimental: Number of times event occurs ÷ Total trials
- Theoretical assumes equally likely outcomes.
5. What is a sample space in theoretical probability?
A sample space is the complete set of all possible outcomes of a random experiment.
- Denoted by S.
- Example (coin toss): S = {H, T}
- Example (die roll): S = {1,2,3,4,5,6}
6. What are equally likely outcomes in probability?
Equally likely outcomes are outcomes that have the same chance of occurring in a random experiment.
- Example: Each face of a fair die has probability 1/6.
- Example: Heads and tails in a fair coin each have probability 1/2.
7. Can theoretical probability be greater than 1?
No, theoretical probability cannot be greater than 1 because probability values range from 0 to 1.
- 0 means the event is impossible.
- 1 means the event is certain.
- All probabilities satisfy 0 ≤ P(E) ≤ 1.
8. What is the probability of an impossible and certain event?
The probability of an impossible event is 0, and the probability of a certain event is 1.
- Impossible event: Rolling a 7 on a standard die → P = 0
- Certain event: Getting a number less than 7 on a die → P = 1
9. How do you find the probability of multiple events in theoretical probability?
For multiple independent events, multiply their probabilities using P(A and B) = P(A) × P(B).
- Example: Tossing two coins, probability of two heads = 1/2 × 1/2 = 1/4.
- This rule applies to independent events.
10. Can you give a real-life example of theoretical probability?
A real-life example of theoretical probability is calculating the chance of drawing a red card from a deck of 52 cards, which is 26/52 = 1/2.
- Total cards = 52
- Red cards (hearts + diamonds) = 26
- Probability = 26/52 = 1/2
