1. Introduction
The laws of probability help in calculating the likelihood of one or more events occurring together or separately. Two fundamental laws in probability are:
- Addition Law → Used when finding the probability of either one event OR another event occurring.
- Multiplication Law → Used when finding the probability of two or more events occurring together (AND rule).
These laws are widely applied in business decision-making, risk assessment, stock market predictions, quality control, and AI models.
2. Addition Law of Probability
(i) Definition
The Addition Law of Probability states that the probability of at least one of two events occurring is given by:
P(A∪B) = P(A) + P(B) − P(A∩B)
Where:
- P(A∪B) = Probability of either A OR B happening.
- P(A) = Probability of event A occurring.
- P(B) = Probability of event B occurring.
- P(A∩B) = Probability of both A AND B occurring together (overlap).
(ii) Special Cases of the Addition Law
1. Mutually Exclusive Events (No Overlap)
If events A and B are mutually exclusive (they cannot happen together), then:
P(A∩B)=0
So, the formula simplifies to:
P(A∪B) = P(A) + P(B)
Example:
A deck of 52 cards is drawn. What is the probability of drawing a King OR a Queen?
P(K∪Q) = P(K) + P(Q) =4/ 52+ 4/ 52 = 8/ 52=0.1538
✔ Advantage: Simple when events do not overlap.
❌ Limitation: Works only if events cannot happen together.
2. Non-Mutually Exclusive Events (With Overlap)
If events A and B are NOT mutually exclusive, meaning they can happen together, then we subtract the overlap:
P(A∪B) = P(A) + P(B) − P(A∩B)
Example:
A student is selected randomly.
- Probability of being good at Math: P(M) = 0.5
- Probability of being good at Science: P(S) = 0.4
- Probability of being good at both Math and Science: P(M∩S) = 0.2
P(M∪S) = P(M) + P(S) − P(M∩S)
=0.5 + 0.4−0.2 = 0.7
✔ Advantage: Avoids double counting of overlapping probabilities.
❌ Limitation: Requires accurate data on joint probabilities.
3. Multiplication Law of Probability
(i) Definition
The Multiplication Law of Probability is used to find the probability of two or more events happening together (AND condition).
For independent events:
P(A∩B) = P(A) × P(B)
For dependent events (where event B depends on event A):
P(A ∩ B ) = P(A) × P(B∣A)
Where:
- P(A∩B) = Probability of both A AND B occurring.
- P(A) = Probability of event A occurring.
- P(B∣A) = Probability of B occurring given A has already occurred.
(ii) Special Cases of the Multiplication Law
1. Independent Events (Events Do Not Affect Each Other)
Two events A and B are independent if the occurrence of one does not affect the occurrence of the other.
P(A ∩ B) = P(A) × P(B)
Example:
What is the probability of getting heads twice when flipping a fair coin?
P(H∩H) = P(H) × P(H)
=1/ 2×1/ 2 =1/ 4 = 0.25
✔ Advantage: Simple calculation when events do not influence each other.
❌ Limitation: Works only for independent events.
2. Dependent Events (Conditional Probability Applies)
If event B depends on event A happening first, we use conditional probability😛
(A∩B) = P(A) × P(B∣A)
Example:
A box contains 5 red and 3 blue balls. A ball is drawn without replacement, and then another ball is drawn. What is the probability of drawing two red balls?
1st draw: P(R1)=5/ 8
2nd draw (since one red ball is removed): P (R2∣R1)=4/ 7
P(R1∩R2) = P(R1) × P(R2∣R1)
=5/ 8×4/ 7=20/ 56=0.357
✔ Advantage: Useful when events depend on each other (e.g., quality control, customer behavior).
❌ Limitation: Requires conditional probabilities, which can be complex.
4. Comparison of Addition and Multiplication Laws
5. Applications of Addition and Multiplication Laws
(i) Business Decision-Making
- Addition Rule: Finding the probability of at least one product defect in quality testing.
- Multiplication Rule: Estimating the probability of multiple equipment failures happening together.
(ii) Stock Market & Finance
- Addition Rule: Calculating the probability of stock prices increasing in at least one sector.
- Multiplication Rule: Analyzing risk of multiple investments failing together.
(iii) Healthcare & Medicine
- Addition Rule: Probability of a patient having either flu OR cold.
- Multiplication Rule: Probability of a patient testing positive for multiple diseases.
(iv) Artificial Intelligence & Machine Learning
- Addition Rule: Probability of a customer clicking on at least one advertisement.
- Multiplication Rule: Probability of multiple AI models failing together.
6. Conclusion
- Addition Law calculates the probability of either event happening.
- Multiplication Law calculates the probability of both events happening together.
- Both laws are widely used in business, finance, AI, and risk analysis.