You must have heard the term “probability” being coined for predicting the weather forecast in news TV bulletins for the next few days for some parts of the country. For calculating the probability of different types of situations, the probability formula and its related basic concepts are used. Probability is the way to measure the uncertainty, how likely an event has happened or is bound to happen.
Probability is that branch of mathematics that is concerned with the numerical description of how likely there are chances of the event to occur or how likely a particular proposition is true. For any event the probability lies between 0 to 1. 0 indicates the impossibility of the event to happen while 1 indicates certainty that the event is certain to occur.
Notation of Probability
For example: let us consider that two events are taking place namely A and B. So for the probability that event A can happen, we are going to write P(A) and for the probability that event B can happen, we can write P(B).
Terminologies Related to Probability Formula
There are a few crucial terminologies that are associated with all probability formulas
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Outcome: The result of an event after experimenting with the side of the coin after flipping, the number appearing on dice after rolling and a card is drawn out from a pack of well-shuffled cards, etc.
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Event: The combination of all possible outcomes of an experiment like getting head or tail on a tossed coin, getting an even or odd number on dice, etc.
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Sample Space: The set of all possible results or outcomes. This indicates that besides this there is no chance that any other result will come. Whatever the result is, it is from this sample Space only.
Probability Formulas with Examples
To Calculate the probability of an event to occur we use this probability formula, recalling, the probability is the likelihood of an event to happen. This formula is going to help you to get the probability of any particular event.
This formula is the number of favourable outcomes to the total number of all the possible outcomes that we have already decided in the Sample Space.
The probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes)
P(A) = n(E) / n(S)
P(A) < 1
Here, P(A) means finding the probability of an event A, n(E) means the number of favourable outcomes of an event and n(S) means the set of all possible outcomes of an event.
If the probability of occurring an event is P(A) then the probability of not occurring an event is
P(A’) = 1- P(A)
Example 01: Probability of obtaining an odd number on rolling dice for once.
Solution: Sample Space = {1, 2, 3, 4, 5, 6}
n(S) = 6
Favourable outcomes = {1, 3, 5}
n(E) = 3
Using the probability formula,
P(A) = n(E) / n(S)
P(Getting an odd number) = 3 / 6 = ½ = 0.5
Important List of Probability Formulas
You just need to have the events for which you are looking for the probability and the formulas are going to make your work easier. In the formulas given below, we are taking 2 events namely A and B. The formulas are based on these events only.
P (A U B) = P (A) + P (B) – P (A ∩ B)
P (A ∩ B) = P (A) . P (B)
P(A NOT B) = A – B
P(B NOT A) = B – A
Probability of occurrence of an event is P(A)
Probability of non-occurrence of the same event is P(A’).
Some probability important formulas based on them are as follows:
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P(A.A’) = 0
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P(A.B) + P (A’.B’) = 1
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P(A’B) = P(B) – P(A.B)
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P(A.B’) = P(A) – P(A.B)
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P(A+B) = P(AB’) + P(A’B) + P(A.B)
Example 01: Two dice are rolled simultaneously. Calculate the probability of getting the sum of the numbers on the two dice is 6.
Solution:
Sample Space= (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6){(1, 1), (1, 2), (1,3), (1,4), (1,5), (1, 6)} {(2, 1), (2, 2),(2,3), (2,4), (2,5), (2, 6)} {(3, 1), (3, 2), (3,3), (3,4), (3,5), (3, 6)} {(4, 1), (4, 2), (4,3), (4,4), (4,5), (4, 6)} {(5, 1), (5,2), (5,3), (5,4), (5,5), (5, 6)} {(6, 1), (6, 2), (6,3), (6,4), (6,5), (6, 6)} n(S) = 36
Favourable outcomes = {(1, 5), (2, 4), (3, 3), (4, 2) and (5, 1)}
n(E) = 5
Using, P(A) = n(E) / n(S)
P(Getting sum of numbers on two dice 6) = 5/ 36