PROBABILITY
PROBABILITY
✤ It is a chance of success of any event.
Prob. of success
P(A) = NO. of favourable Cases / Total no. of cases.
1. P(A) = n(E) /n(S)
Prob. of failer = 1 - Prob of sucess
P(A’)=1-P(A)
✥ let A and B any two events then Prob of success of A or B
P(AUB) = P(A)+P(B)-P(A∩B)
✥ If A and B any two mutually exclusive event then
P(A∩B) = 0
✥ Prob. of neither A nor R.
P(A∩B) = 1 - P(AUB)
=1-(P(A)+P(B)-P(A∩B))
P(A’UB’) = P(A∩B)’
= 1-P(A∩B)
✥ Prob. of A but not B
P(A∩B’) = P(A-B)
= P(A) - P(A∩B)
✥ whenever we have to select two or more object from 'n given objects,
We should use nCr
✥ Sample space: Collection of all possible outcomes associate with
given experiments.
Ex :- when a fair dice is thrown one time then s = {1,2,3,4,5,6}
✥ Compound event: Collection of two or more event associated
which are used to find Prob.
✥ If AUB = sample space then A & B are called mutually exhaustive event.
✥ If A∩B= φ then A and B are called Mutually exclusive event.
Ex :- A = {1,3,5}
B = {4,4,6,2}
A∩B= φ
✤ Conditional Probability: → let A and B any two events. Then Probability
of A when B already occurs, is called conditional Probability.
It is represented by P(A/B) = P(A∩B)/P(B)
or P(B/A) = n(A∩B)/n(B).
2. P(B/A) = P(A∩B) / P(A)
= n(A∩B) / n(A)
3. If A is subset of B (A ⊂ B)
then P(A/B) = 1.
P(B/B) = 1.
✤ Independent event: let A and & any two events, If occurance of A is not
affected by B and vice-versa then A and B are called independent
event.
for independent P(A∩B) = P(A) . P(B).
2. Prob. of A or B, P(A∪B) = P(A) + P(B) - P(A) . P(B).
3. P(A/B) = P(A) P(B/A) = P(B)
4. Prob. of neither A nor B → P(Ā∩B) = P(A∪B)
= 1 - P(A∪B)
✤ Total probability law: let E₁, E₂, …Eₙ. are n events, A is any event which
occurs with E₁ or E₂ … or Eₙ then.
P(A) = P(E₁) . P(A/E₁) + P(E₂) . P(A/E₂) + … P(Eₙ) . P(A/Eₙ)
✤ Bayes's Theorem: It is the special case of total probability law
P(E₁/A) = P(E₁) . P(A/E₁) / P(E₁) . P(A/E₁) + P(E₂) . P(A/E₂) + …
* * * * * * * * * * * * *
