What is Addition Theorem of Probability for Three Events? | Simple Explanation for Class 12

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What is the Addition Theorem of Probability for Three Events?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The Addition Theorem of Probability for Three Events helps us find the chance that at least one of three events (A, B, or C) will happen. It tells us how to add their individual probabilities while avoiding counting overlaps more than once. This theorem is crucial when events might occur together.

Simple Example

Quick Example

Imagine you are trying to score marks in your Class 12 board exams. Let A be passing in Physics, B be passing in Chemistry, and C be passing in Maths. The Addition Theorem for three events helps you calculate the probability that you pass in at least one of these three subjects.

Worked Example

Step-by-Step

Let's say a student applies for three internships: Internship A, Internship B, and Internship C. The probabilities of getting selected for each are: P(A) = 0.4, P(B) = 0.3, P(C) = 0.2. The probabilities of overlaps are: P(A and B) = 0.1, P(A and C) = 0.08, P(B and C) = 0.05. The probability of getting all three is P(A and B and C) = 0.02. What is the probability of getting at least one internship?

---1. Write down the formula for P(A or B or C):
P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C)

---2. Substitute the given values into the formula:
P(A or B or C) = 0.4 + 0.3 + 0.2 - 0.1 - 0.08 - 0.05 + 0.02

---3. Add the individual probabilities:
0.4 + 0.3 + 0.2 = 0.9

---4. Subtract the probabilities of two events happening together:
0.9 - 0.1 - 0.08 - 0.05 = 0.9 - 0.23 = 0.67

---5. Add back the probability of all three events happening together:
0.67 + 0.02 = 0.69

---Answer: The probability of getting at least one internship is 0.69.

Why It Matters

This theorem is vital in AI/ML for understanding complex system failures, in FinTech for risk assessment of multiple investments, and in Medicine for calculating the chances of a patient having at least one of several conditions. Engineers use it to design reliable systems, and data scientists rely on it to make predictions and build models.

Common Mistakes

MISTAKE: Simply adding P(A) + P(B) + P(C) without accounting for overlaps. | CORRECTION: Remember that simply adding individual probabilities counts the overlapping parts (where two or three events happen together) multiple times. You must subtract these overlaps.

MISTAKE: Forgetting to add back P(A and B and C) at the end. | CORRECTION: After subtracting the two-event overlaps, the region where all three events occur together (A and B and C) gets subtracted three times and added once. So, it is effectively subtracted twice. You must add it back one more time to count it correctly.

MISTAKE: Confusing 'A or B or C' with 'A and B and C'. | CORRECTION: 'A or B or C' means at least one of the events happens. 'A and B and C' means all three events happen simultaneously. The Addition Theorem calculates 'or'.

Practice Questions

Try It Yourself

QUESTION: If P(X)=0.5, P(Y)=0.4, P(Z)=0.3, P(X and Y)=0.2, P(X and Z)=0.15, P(Y and Z)=0.1, and P(X and Y and Z)=0.05, find P(X or Y or Z). | ANSWER: P(X or Y or Z) = 0.5 + 0.4 + 0.3 - 0.2 - 0.15 - 0.1 + 0.05 = 1.2 - 0.45 + 0.05 = 0.75 + 0.05 = 0.8

QUESTION: In a market survey, 60% like Chai (C), 50% like Coffee (F), and 40% like Lassi (L). 30% like C and F, 20% like C and L, 10% like F and L. If 5% like all three, what percentage like at least one of these drinks? | ANSWER: P(C or F or L) = 0.60 + 0.50 + 0.40 - 0.30 - 0.20 - 0.10 + 0.05 = 1.50 - 0.60 + 0.05 = 0.90 + 0.05 = 0.95 or 95%

QUESTION: Three friends, Ram, Shyam, and Geeta, are trying to get a top rank in their school. The probability of Ram getting a top rank is 0.7, Shyam is 0.6, and Geeta is 0.5. The probability of Ram and Shyam both getting a top rank is 0.4. Ram and Geeta is 0.3. Shyam and Geeta is 0.2. The probability of all three getting a top rank is 0.1. What is the probability that at least one of them gets a top rank? | ANSWER: P(R or S or G) = P(R) + P(S) + P(G) - P(R and S) - P(R and G) - P(S and G) + P(R and S and G) = 0.7 + 0.6 + 0.5 - 0.4 - 0.3 - 0.2 + 0.1 = 1.8 - 0.9 + 0.1 = 0.9 + 0.1 = 1.0

MCQ

Quick Quiz

Which term is added back in the Addition Theorem for three events after subtracting all two-event overlaps?

The probability of the first event only

The probability of all three events happening together

The probability of the second event only

The sum of all individual probabilities

The Correct Answer Is:

The term P(A and B and C) is added back because it was subtracted three times when removing the two-event overlaps, but it was initially included once in each of P(A), P(B), and P(C). Adding it back ensures it is counted exactly once.

Real World Connection

In the Real World

In cricket analytics, this theorem helps teams understand the probability of at least one of three key batsmen scoring a half-century in a match. Data analysts at sports companies like Star Sports use this to predict game outcomes and player performance, influencing strategies and fantasy league predictions.

Key Vocabulary

Key Terms

Probability: The chance of an event happening, usually a number between 0 and 1. | Event: A specific outcome or set of outcomes in an experiment. | Overlap: When two or more events occur at the same time. | Union (OR): The event that at least one of the specified events occurs. | Intersection (AND): The event that all specified events occur simultaneously.

What's Next

What to Learn Next

Next, you can explore Conditional Probability and Bayes' Theorem. These concepts build on the Addition Theorem by helping you understand how the probability of an event changes if you already know another event has happened. This is super useful for more advanced problem-solving!