Class 11: Probability
Events, Axiomatic Approach & Algebra of Events
1. Basic Concepts & Definitions
| Event ($$E$$) | Any subset $$E$$ of a sample space$$S$$ is called an event. |
| Occurrence of Event | An event $$E$$ is said to have occurred if the outcome $$\omega$$ of the experiment satisfies $$\omega \in E$$. |
| Impossible Event | The empty set $$\phi$$ is called an impossible event ($$P(\phi) = 0$$). |
| Sure Event | The whole sample space $$S$$ is called the sure event ($$P(S) = 1$$). |
| Simple (Elementary) Event | An event having only one sample point of a sample space. |
| Compound Event | An event that has more than one sample point. |
2. Algebra of Events
| Complementary Event (“Not A”) | $$A’ = S – A = \{ \omega : \omega \in S \text{ and } \omega \notin A \}$$. |
| Event “A or B” (Union) | $$A \cup B = \{ \omega : \omega \in A \text{ or } \omega \in B \}$$. |
| Event “A and B” (Intersection) | $$A \cap B = \{ \omega : \omega \in A \text{ and } \omega \in B \}$$. |
| Event “A but not B” (Difference) | $$A – B = A \cap B’$$. |
| Mutually Exclusive Events | Two events $$A$$and$$B$$are mutually exclusive if they cannot occur simultaneously, i.e.,$$A \cap B = \phi$$. |
| Exhaustive Events | [cite_start]Events $$E_1, E_2, \dots, E_n$$are exhaustive if their union is the sample space:$$\bigcup_{i=1}^{n} E_i = S$$. |
3. Axiomatic Approach to Probability
Let $$S = \{ \omega_1, \omega_2, \dots, \omega_n \}$$ be the sample space.
| Axioms | 1. $$0 \le P(\omega_i) \le 1$$ for each outcome. 2. $$\sum P(\omega_i) = 1$$. 3. For any event $$A$$, $$P(A) = \sum_{\omega_i \in A} P(\omega_i)$$. |
| Equally Likely Outcomes | If all outcomes are equally likely, $$P(\omega_i) = \frac{1}{n}$$. For an event $$E$$: $$P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of outcomes favorable to E}}{\text{Total possible outcomes}}$$. |
4. Probability Formulas
| Addition Theorem (P(A or B)) | $$P(A \cup B) = P(A) + P(B) – P(A \cap B)$$. |
| Mutually Exclusive Events | If $$A \cap B = \phi$$, then $$P(A \cup B) = P(A) + P(B)$$. |
| Probability of “Not A” | $$P(A’) = 1 – P(A)$$. |
| Union of 3 Events | $$P(A \cup B \cup C) = P(A) + P(B) + P(C) – P(A \cap B) – P(B \cap C) – P(A \cap C) + P(A \cap B \cap C)$$. |