1.8 Basic Probability¶
Key Concepts:¶
Probability:¶
- Numerical measure of the likelihood of an event.
- Ranges from 0 to 1:
- 0: Event is impossible.
- 1: Event is certain.
- 0.5: Event is equally likely or unlikely.
Random Experiment:¶
- A process with more than one possible outcome.
- Example: Revenue outcomes for movies KGF (3, 4, 5, 6 lakhs) and Kantara (4, 5, 6 lakhs).
Sample Space (S):¶
- Set of all possible outcomes of a random experiment.
- Example: Combining outcomes for KGF and Kantara yields 12 outcomes in the sample space.
Event (A):¶
- A subset of the sample space, representing outcomes of interest.
- Example: Event A (KGF is a hit) includes outcomes where revenue is 5 or 6 lakhs.
Complement of an Event (A'):¶
- Consists of all outcomes in the sample space not in event A.
- Example: If A = {KGF is a hit}, then A' = {KGF is not a hit}.
Rules of Probability:¶
Assignment:¶
- Probability of any outcome is between 0 and 1.
- The sum of probabilities of all outcomes in the sample space is 1.
Calculating Event Probability:¶
- Sum the probabilities of all outcomes within the event.
- Example:
- Event A: KGF is a hit (revenue ≥ 5 lakhs).
- Outcomes: {5, 6 lakhs}.
- Probabilities: 0.1 + 0.08 + … = 0.34.
- Event B: Kantara is a hit (revenue ≥ 5 lakhs).
- Outcomes: {5, 6 lakhs}.
- Probabilities: 0.1 + 0.1 + … = 0.6.
Complement Rule:¶
- \( P(A') = 1 - P(A) \).
- Example:
- For Event A: \( P(A) = 0.34 \) → \( P(A') = 1 - 0.34 = 0.66 \).
- For Event B: \( P(B) = 0.6 \) → \( P(B') = 1 - 0.6 = 0.4 \).
Application Example:¶
KGF Revenue:¶
- Possible values: 3, 4, 5, 6 lakhs.
- Probability KGF is a hit (≥ 5 lakhs): 0.34.
- Complement (not a hit): 0.66.
Kantara Revenue:¶
- Possible values: 4, 5, 6 lakhs.
- Probability Kantara is a hit (≥ 5 lakhs): 0.6.
- Complement (not a hit): 0.4.
Visualization:¶
Venn Diagrams:¶
- Sample space represented as a rectangle.
- Events represented as circles.
- Complements shown as the area outside the circle but inside the rectangle.