# Axiomatic Approach to Probability

• Probability : Number P(ωi) associated with sample point ωi such that
(i) 0 ≤ P (ωi) ≤ 1
(ii) Σ P(ωi) for all ωi ∈ S = 1
(iii) P(A) = Σ P(ωi) for all ωi ∈ A. The number P(ωi) is called probability of the outcome ωi
• Equally likely outcomes : All outcomes with equal probability
• Probability of an event: For a finite sample space with equally likely outcomes
Probability of an event P(A) = \frac{n(A)}{n(S)}, where n(A) = number of elements in the set A, n(S) = number of elements in the set S.
• If A and B any two events, then
P(A or B) = P(A) + P(B) − P(A and B)
equivalently, P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• If A and B are mutually exclusive, then P(A or B) = P(A) + P(B)
• If A is any event, then
P(not A) = 1 − P(A)
• If n letter corresponding to n envelopes are placed in the envelopes at random, then
(i) Probability that all letters are in right envelopes = 1/n!.
(ii) Probability that all letter are not in right envelopes = 1 - \frac{1}{n!}.
(iii) Probability that no letter is in right envelopes = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - .... + (-1)^{n} \frac{1}{n!}
Probability that exactly r letters are in right envelopes = \frac{1}{r!}\left[\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - .... + (-1)^{n-r} \frac{1}{(n-r)!}\right]
• If odds in favour of an event are a : b, then the probability of the occurrence of that event is \frac{a}{a + b} and the probability of non-occurrence of that event is \frac{b}{a + b}.
• If odds against an event are a : b, then the probability of the occurrence of that event is \frac{b}{a + b} and the probability of non-occurrence of that event is \frac{a}{a + b}.
• Let A, B, and C are three arbitrary events. Then
(i) Only A occurs is A \cap \overline{B} \cap \overline{C}
(ii) Both A and B, but not C occur is A \cap B \cap \overline{C}
(iii) All the three events occur is A ∩ B ∩ C
(iv) At least one occurs is A ∪ B ∪ C
(v) At least two occur is (A ∩ B) ∪ (B ∩ C) ∪ (A ∩ C)
(vi) None occurs is \overline{A} \cap \overline{B} \cap \overline{C} = \overline{A \cup B \cup C}
(vii) Exactly one of A and B occurs is \left(A \cap \overline{B}\right) \cup \left(\overline{A} \cap B\right)
• P (exactly one of them out of three events A, B and C)
\tt =P(A \cap \overline{B} \cap \overline{C}) + P(\overline{A} \cap B \cap \overline{C}) + P(\overline{A} \cap \overline{B} \cap C)
• P (exactly two of them out of three events A, B and C)
\tt =P(A \cap B \cap \overline{C}) + P(A \cap \overline{B} \cap C) + P(\overline{A} \cap B \cap C)
• \tt P(A \cap \overline{B}) + P(\overline{A} \cap B) = P(A) + P(B) - 2P(A \cap B) = P(A ∪ B) − P(A ∩ B)
• If A and B are any two events, then
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A) + P(B) − P(A ∪ B)
• If A, B, C are any three events, then P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C)
• Demorgan's laws
\tt P(\overline{A \cup B}) = P(\overline{A} \cap \overline{B})
\tt P(\overline{A \cap B}) = P(\overline{A} \cup \overline{B})
• The probability of odds favarable to an event A is denoted by \tt P(A) : P(\overline{A})
• The probability of odds against to an event A is denoted by \tt P(\overline{A}) : P(A)
• If P(A) is the probability of occurrence of an event A, then the probability of nonoccurrence of an event A is denoted by \tt P(\overline{A}).
\tt \therefore P(A) + P(\overline{A}) = 1
• If A is Null event (or) Impossible event then P(A) = 0
• If A is sure event (or) certain event, then P(A) = 1
• If A and B are mutually exclusive events, then P(A ∩ B) = 0
P(A ∪ B) = P(A) + P(B)
• If A and B are exhaustive events, then P(A ∪ B) = 1
• If A, B, C are any three events, then
0 ≤ P(A ∪ B) ≤ 1
0 ≤ P(A ∪ B ∪ C) ≤ 1
P(A) ≤ P(A ∪ B), P(B) ≤ P(A ∪ B)
P(A ∩ B) ≤ P(A), P(A ∩ B) ≤ P(B)

### View the Topic in this video From 1:06 To 14:50

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1. (i) P (A ∪ A') = S
(ii) P (A ∩ A') = Φ
(iii) P (A')' = A
2. If a set of events A1, A2, ....., An are mutually exclusive, then A1 ∩ A2 ∩ A3 ∩..... ∩ An = Φ
∴ P(A1 ∪ A2 ∪ A3 ∪ ..... ∪ An) = P(A1) + (A2) + .... + P(An) and P(A1 ∩ A2 ∩ A3 ∩..... ∩ An) = 0
3. If a set of events, A1, A2, ..... An are exhaustive, then P(A1 ∪ A2 ∪ ..... ∪ An) = 1
4. Probability of any event in a sample space is 1. i.e., P(A) = 1
5. Odds in favour of A=\frac{P(A)}{P(\overline{A})}
6. Odds in against of A=\frac{P(\overline{A})}{P(A)}
7. For two events A and B
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
8. For three events A, B and C
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C)
9. P(\bar{A} \cap B)=P(B)-P(A \cap B)
10. P(A \cap \bar{B})=P(A)-P(A \cap B)
11. P[(A \cap \bar{B}) \cup(\bar{A}\cap B)]=P(A)+P(B)-2P(A \cap B)
12. P(\bar{A} \cap \bar{B})=1-P(A \cup B)
13. P(\bar{A} \cup \bar{B})=1-P(A \cap B)
14. P(A) = P(A \cap B)+P(A \cap \bar{B})
15. P(B) = P(A \cap B)+P(B \cap \bar{A})
16. P(\bar{A}) = 1-P(A)
17. P(A\cup \bar{A}) = P(S)=1, P(\phi)=0
18. P(A ∩ B) = P(A) × P(B), if A and B are independent events.