[Maths Class Notes] on Theoretical Probability Pdf for Exam

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

 

Uploaded soon)

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 outcome
s.

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.