Probability- A Theoretical Approach: An Overview and Examples

Probability is the calculation of the expected particular event. In Mathematics, the probability is expressed between the range of 0 and 1 (0 represents the impossibility of an event and 1 indicates the certainty).

There are three fundamental rules for probability: addition, multiplication, and complement rules. In General, whenever we are not sure about a certain future event, we can guess and talk about the probable event that is uncertain in nature. We have three types of probability: 

• Theoretical probability
• Experimental probability
• Axiomatic probability.

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Theoretical Probability

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Theoretical probability is the theoretical approach behind probability. For this particular approach, the practical experiment is not required to find the probable occurrences. Instead, we should be aware of the probability situation of the uncertain event. This theoretical approach of probability is defined as the difference between the number of favorable outcomes and the no. of possible outcomes.

Probability of Event P (E) = No. of. Favorable outcomes/ No. of. Possible outcomes

Example

Question: Find the probability of rolling dice, when it rolls a 4

Answer: To find this we don’t need any experiment, as we know that rolling dice has 6 possible outcomes 1,2,3,4,5,6. Therefore the probability is,

P(E)=No. of favorable outcomes/No. of possible outcomes

P(E)= 1/6

Hence, the probability to get 4 while rolling dice is 1/6.

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Experimental Probability

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Experimental probability is also known as empirical probability. Experimental probability is computed on the basis of the actual outcomes of an uncertain event. It is the difference between the No. of times that event occurs and the Total No. of Trials of the experiment.

Probability of Event P(E) = No. of Times that event occurs/ Total number of trials

Example

Question: Tim and Tom tossed a coin 10 times in a row. The probable outcomes of this experiment are as follows:

Find the experimental probability for each outcome.

Answer:

Formula, P (Occurrence of Heads) = No. of times head occurs/Total No. of Trails

P (Occurrence of Tails) = No. of times tails occurs/Total No. of Trails

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Theoretical Probability and Experimental Probability: Comparison

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Before comparing both the theoretical and experimental approach of probability, we should be clear with the definitions and fundamental aspects. 

• Firstly, in the case of experimental probability, we conduct the experiment continuously to know the random outcomes and to compute the probability of the outcomes. These experiments are known as random experiments and these outcomes are unpredictable. The collection of probable outcome is what form up an event. Each repetition for seeing probability is called a trail.

The probability of an event is:

P(E)= The count of favorable outcomes/Total No. of possible outcomes

In experimental probability, the No. of trails is so high at this time experimental probability gets into theoretical probability. 

• The theoretical approach utilizes the knowledge of uncertain probable outcomes and it doesn’t have any experiment.

The experiment of tossing a coin or rolling dice can be repeated many times to get a probable outcome. But in real-life situations such as finding the probability of launching a satellite experiment cannot be done multiple times because launching a satellite using a missile multiple times is neither practical nor possible. In such type of real-life cases, it becomes very important to make certain assumptions and based on these assumptions theoretical probability comes into the frame that is very useful in such practical cases, especially in a lot of circumstances where we cannot perform the experiment.

Probability: The Complement Rule

We have learned how to calculate a probability for an event that is possible. But sometimes we are tempted to find the probability for an event that will not ever happen. Here the complement rule of probability comes into the picture. The probability that the complement of an event will occur is denoted by: 

P(E’)=1−P(E)