Probability theorems
There are different types of probability theorems they are used in many cases i.e. they are Normal probability theorem, Binomial probability theorem, Baye’s theorem of probability.
Binomial probability theorem
Binomial probability is used to calculate individual and cumulative probabilities. A Binomial distribution is one of the probability distribution. It refers to the probabilities associated with the number of successes in a Binomial Experiment. Binomial probability is used for calculating or creating new binomial probability problems.
Binomial probability is determined using the formula
Example: What is the probability of getting exactly 2 Heads when 3 coin tosses?
Given that
p = 1/ 2
q = 1 – p = 1- 1/ 2 = 1/ 2
k = 2, n = 3
Binomial probability:
P(X=2)=3/2(1/2)2(1/2)3-2
= 3 x (1/ 4) x (1/ 2)
= 3 x 1/ 8 = 3/8
= 0.375
The probability of getting exactly 2 Heads when 3 coin tosses is 0.375.
Normal Probability Theorem
Normal probability is general probability i.e. used to calculate the normal distribution variables. It shows us the number of occurrence of an event out of possible outcomes.
Normal probability ‘P’ = Number of possible outcomes / Sample space.
Example:
A bag contains 6 black balls, 4 blue balls and 3 grey balls; a ball is picked up at random. Find the probability to select a black ball.
Given that
Black balls = 6
Blue balls = 4
Grey balls = 3
Total number of balls = 6 + 4 + 3 = 13
Sample space =13!/(13-1)!
= 13
Number of possibilities =6!/(6-1)!
= 6
Normal probability ‘P’ = Number of possible outcomes / Sample space.
P(B) = 6/13, Therefore probability is 6/13.
Baye’s Theorem
Baye's theorem shows the relation between one conditional probability and its inverse; consider two conditional probabilities like A and B the probability of event A given event B depends not only on the relationship between A and B but also on the absolute probability (occurrence) of A not concerning B, and the absolute probability of B not concerning A. It is generally represented using following formula
Using all these above formulas we can determine the probability using Bayes’ theorem.
Example:
Find the relation between the conditional probabilities using Baye’s theorem. Whereas the probabilities are 3/ 4 and 2/4
Given that 9441877172
P(A) = 3/ 4
P(B) = 2/4
Baye’s theorem:
P(B?A) = P(A?B)/P(A)
P(A?B) = 3/ 4 – 2/4 = ¼
P(B?A) = P(A?B)/P(A)
= 1/ 4 / 3/ 4 = 1/3.
P(A?B) = P(A?B)/P(B)
= 1/ 4 / 2/4 = 1/ 2.
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