How to Calculate Theoretical Probability in Simple Steps
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: Definition, Formula & Solved Examples
1. What is theoretical probability in simple terms?
Theoretical probability is a way of calculating the chance of an event happening based on reasoning and logic, without performing any experiments. It assumes that all possible outcomes are equally likely. For example, when you toss a fair coin, you reason that there are two equally likely outcomes (Heads or Tails), so the theoretical probability of getting Heads is 1 out of 2, or 1/2.
2. What is the official formula for calculating theoretical probability?
The formula for calculating the theoretical probability of an event (often written as P(E)) is:
P(E) = (Number of Favourable Outcomes) / (Total Number of Possible Outcomes).
A 'favourable outcome' is the specific result you are interested in, while the 'total number of possible outcomes' represents all the possible results that could occur in the experiment (also known as the sample space).
3. How does theoretical probability differ from experimental probability?
The main difference lies in how they are determined:
Theoretical Probability is based on logic and what we expect to happen. It's calculated before any trial, like saying the probability of rolling a 4 on a standard die is 1/6.
Experimental Probability is based on the results of an actual experiment. If you roll a die 100 times and get a 4 twenty times, the experimental probability is 20/100 or 1/5. It is what actually happens in practice.
As the number of trials in an experiment increases, the experimental probability often gets closer to the theoretical probability.
4. Can you provide a real-life example of using theoretical probability?
A common real-life example of theoretical probability is in games of chance. For instance, in a standard deck of 52 playing cards, if you want to find the probability of drawing a King, you use theoretical probability. You know there are 4 Kings (favourable outcomes) and 52 total cards (total possible outcomes). Therefore, the theoretical probability is 4/52, which simplifies to 1/13. This calculation is done without actually drawing any cards.
5. What are the basic steps to solve a theoretical probability problem?
To solve a theoretical probability problem, you can follow these steps:
Step 1: Identify the experiment or action (e.g., rolling a die, picking a card).
Step 2: Determine the total number of all possible outcomes. This is the size of your sample space.
Step 3: Identify the specific event you are interested in and count the number of 'favourable' outcomes for that event.
Step 4: Apply the formula: Divide the number of favourable outcomes by the total number of possible outcomes.
6. Why is it called 'theoretical' if we use it for real-world calculations?
It is called 'theoretical' because it's based on an idealised model of a situation, not on actual results. We assume perfect conditions, such as a perfectly fair coin or a completely random shuffle of cards. In the real world, minor imperfections might exist. However, this theoretical model is extremely powerful because it provides a reliable baseline or prediction for what should happen on average, making it a fundamental tool in fields like statistics, science, and finance.
7. In what situations might theoretical probability not accurately predict an outcome?
Theoretical probability works best when all outcomes are truly equally likely. It may not be an accurate predictor in situations where there is bias. For example, if you are using a loaded die (weighted to favour a certain number), the theoretical probability of 1/6 for each face is incorrect. Similarly, in complex real-world events like predicting the weather or stock market prices, outcomes are not equally likely and are influenced by numerous factors, making theoretical probability an unsuitable model on its own.
8. What does a theoretical probability of 0 or 1 signify?
A theoretical probability of 0 or 1 represents the two extremes of certainty:
A probability of 0 means the event is impossible. For example, the probability of rolling a 7 on a standard six-sided die is 0, as there are no favourable outcomes.
A probability of 1 means the event is certain to happen. For instance, the probability of rolling a number less than 7 on a standard six-sided die is 6/6 = 1, because all possible outcomes are favourable.
9. How are the 'odds' of an event different from its 'probability'?
While related, odds and probability describe chances in different ways. Probability compares favourable outcomes to the total outcomes, while odds compare favourable outcomes to unfavourable outcomes. For example, the probability of rolling a 2 on a die is 1/6. However, the odds of rolling a 2 are calculated as (1 favourable outcome) to (5 unfavourable outcomes), which is expressed as 1:5 or 1 to 5.
