Probability

Terminology

Experiment: An operation or a trial done to produce all possible outcomes is called an Experiment.

Sample Space:  Sample space is a set of all possible outcomes of an experiment.

Favourable Outcomes: All the outcomes which favour an event, i.e., satisfy the conditions of an event, are called favourable outcomes.

Complement of an Event (A’/ A^c ): Event B is said to be Complement (A’) of Event A if and only if B consists of all the outcomes in which event A doesn’t occur.

Equally Likely Events: Two events are equally likely if the chances of occurrence or probability of occurring of the two events are the same or equal.

Sure Event: If the chances of occurrence of an event are 1, i.e., P(A)=1, or it always occurs whenever the event is performed, the event is called a sure event.

Impossible Event: If the chances of occurrence of an event are 0, i.e., P(A)=0, or it never occurs whenever the event is performed, the event is called an impossible event.

Exhaustive Events: Two events are said to be exhaustive if the union of these events is the sample space. 

Mutually Exclusive Events: Two events are said to be mutually exclusive if their intersection is zero, i.e., they do not occur at the same time or simultaneously.

Formula for Probability

Let A denotes the occurrence of an event, and P(A) represents the probability of the event A. 

P(A) = No. of favourable outcomes(X) / Total number of outcomes(N)

Let’s understand the formula better with an example: Sam has a dice, i.e., a six-faced cube numbered from 1 to 6 on the face. Upon rolling the dice, he was curious to know the chances of getting a number greater than or equal to 3 on the top face. Let’s define an event A that results in a number ≥ 3. 

Possible outcomes for a roll of dice (S) = {1,2,3,4,5,6}

Total number of outcomes(N) = 6

Favourable Outcomes = {3,4,5,6}

Number of Favourable Outcomes(X) = 4

P(A)=No. of favourable outcomes(X) / Total number of outcomes(N)  = 4/6= 2/3

So, the probability of event A is 6.667. 

Theorems

Let A, B, C are the events associated with a random experiment, then

1. P(A∪B) = P(A) + P(B) – P(A∩B)
2. P(A∪B) = P(A) + P(B) if A and B are mutually exclusive
3. P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C)- P(C∩A) + P(A∩B∩C)
4. P(A∩B’) = P(A) – P(A∩B)
5. P(A’∩B) = P(B) – P(A∩B)

Total Probability Theory

Let E1, E2, E3,............, En be n mutually exclusive and exhaustive events of a random experiment. Let S be the sample space. Suppose A is an event occurring with E1 or E2 or.......En, then 

P(A) = P(E1).P(A/E1) + P(E2).P(A/E2) + ... +  P(En).P(A/En)

FAQs

1. What is probability in Mathematics? Give an example.
   The possibility of occurrence of an event is called probability. It is the ratio of favourable outcomes to the total number of outcomes.
   For example, Tossing of a coin leads to two possible outcomes, getting a head or a tail. The probabilities of getting a head or a tail are ½.
    
2. What is the formula for finding the probability of an event?
   Let A denotes the occurrence of an event, and P(A) represents the probability of the event A. 
   Then, P(A)=No. of favourable outcomes(X) / Total number of outcomes(N) 
    
3. Write the inclusion-exclusion equation for probability.
       If A and B are two events, then, P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B )

Key Takeaways

In this article, we have extensively discussed the concepts of probability, terminologies associated and the theorems. We hope that this blog has helped you enhance your knowledge, and if you wish to learn more, check out our playlist Basic Mathematics. You can also go through our Coding Ninjas Blog site and visit our Library. Do upvote our blog to help other ninjas grow.

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