Definitions [2]
The conditional probability of both events A and B over the sample space S is
\[\mathrm{P(A/B)=\frac{P(A\cap B)}{P(B)}}\], where \[B\neq\phi\]
\[\mathrm{P(B/A)=\frac{P(A\cap B)}{P(A)}}\], where \[A\neq\phi\]
Two events are said to be independent if the occurrence of one does not depend on the other.
If A and B are independent events, then
- P(A/B) = P(A/B') = P(A)
- P(B/A) = P(B/A') = P(B)
- If A and B are independent events, then
a. P(A∩ B) = P(A). P (B)
b. A and B' are also independent
c. A' and B' are also independent
Theorems and Laws [3]
If A and B are any two events defined over a sample space S, then P(A∪B) = P(A) + P(B) – P(A∩В)
or
P (A+B) = P(A) + P(B) – P(AB)
Mutually Exclusive Events:
If P(A ∩ B) = 0, then
P(A ∪ B) = P(A) + P(B)
If A and B are two events over the sample space S, then
- P(A ∩ B) = P(B) · P (A/B)
- P(A ∩ B) = P(A) · P (B/A)
If B1, B2,..., Bn are mutually exclusive and exhaustive events and if A is an event consequent to these Bi's, then for each i = 1, 2, 3, ..., n,
\[\mathrm{P(B_i/A)=\frac{P(B_i)P(A/B_i)}{\sum_{i=1}^nP(A\cap B_i)}}\]
Key Points
| Type of Event | Meaning | Probability |
|---|---|---|
| Sure (Certain) Event | An event that is certain to occur | P(E) = 1 |
| Impossible Event | An event that cannot occur | P(E) = 0 |
| Simple (Elementary) Event | An event having only one outcome | P(E) = 1 / n(S) |
| Complementary Event (E̅) | An event that occurs when E does not occur | P(not E) = 1 − P(E) |
| Mutually Exclusive Events | Two events that cannot occur together | P(A ∩ B) = 0 |
| Exhaustive Events | Events which together cover all outcomes of S | P(A₁) + P(A₂) + … = 1 |
| Equally Likely Events | All outcomes have the same chance of occurring | P(E) = n(E) / n(S) |
| General Rule | Probability of any event | 0 ≤ P(E) ≤ 1 |
Properties:
- Complement Rule
P(A′) = 1 − P(A)
⇒ P(A) + P(A′) = 1 - Range of Probability
0 ≤ P(A) ≤ 1 - Impossible Event
P(ϕ) = 0 - Certain Event
P(S) = 1 - Subset Rule
If A ⊆ B, then P(A) ≤ P(B) - Difference of Events
P(A ∩ B′) = P(A) − P(A ∩ B)
P(A′ ∩ B) = P(B) − P(A ∩ B) - Union of Two Events
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) - Union of Three Events
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
− P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C) - Mutually Exclusive Events (2 events)
If A ∩ B = 0, then
P(A ∪ B) = P(A) + P(B) - Mutually Exclusive Events (multiple)
P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ) - Upper Bound of Union
P(A ∪ B) ≤ P(A) + P(B)
- Odds in favour of event A
Odds in favour of A = P(A) / P(A′) - Odds against event A
Odds against A = P(A′) / P(A)
