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What is the Addition Theorem of Probability for Non-Mutually Exclusive Events?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Addition Theorem of Probability for Non-Mutually Exclusive Events helps us find the probability that at least one of two events will happen, even if they can both happen at the same time. It tells us to add the individual probabilities of each event and then subtract the probability of both events happening together, because we counted that part twice.
Simple Example
Quick Example
Imagine you are picking a random student from your class. What is the probability that the student plays cricket OR plays football? If some students play both, you need this theorem. You'd add the probability of playing cricket to the probability of playing football, then subtract the probability of playing both.
Worked Example
Step-by-Step
Let's say in a class of 50 students, 20 like Mango juice, 15 like Orange juice, and 5 students like BOTH Mango and Orange juice. What is the probability that a randomly chosen student likes Mango OR Orange juice?
Step 1: Find the probability of liking Mango juice (Event M).
P(M) = (Number of students liking Mango) / (Total students) = 20/50 = 0.4
---Step 2: Find the probability of liking Orange juice (Event O).
P(O) = (Number of students liking Orange) / (Total students) = 15/50 = 0.3
---Step 3: Find the probability of liking BOTH Mango AND Orange juice (Event M and O).
P(M and O) = (Number of students liking both) / (Total students) = 5/50 = 0.1
---Step 4: Apply the Addition Theorem formula: P(M or O) = P(M) + P(O) - P(M and O).
P(M or O) = 0.4 + 0.3 - 0.1
---Step 5: Calculate the final probability.
P(M or O) = 0.7 - 0.1 = 0.6
Answer: The probability that a randomly chosen student likes Mango OR Orange juice is 0.6 or 60%.
Why It Matters
This theorem is super important for understanding how different events combine. Data scientists use it to predict user behavior on apps, and engineers use it to calculate the reliability of systems. Even in AI/ML, it helps in making smarter decisions by combining probabilities from different sources.
Common Mistakes
MISTAKE: Students often forget to subtract the probability of both events happening. They just add P(A) + P(B). | CORRECTION: Remember that the 'both' part is counted twice when you add P(A) and P(B) separately, so you must subtract P(A and B) once.
MISTAKE: Confusing 'mutually exclusive' with 'non-mutually exclusive' events. | CORRECTION: If events are mutually exclusive (cannot happen together), P(A and B) is 0, and the formula simplifies to P(A) + P(B). For non-mutually exclusive events (can happen together), P(A and B) is NOT 0.
MISTAKE: Incorrectly calculating P(A and B). | CORRECTION: P(A and B) is the probability of the overlap, where both events occur. Make sure you correctly identify and calculate this overlap based on the problem statement.
Practice Questions
Try It Yourself
QUESTION: In a deck of 52 cards, what is the probability of drawing a King OR a Heart? | ANSWER: P(King) = 4/52, P(Heart) = 13/52, P(King and Heart) = 1/52. So, P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.
QUESTION: A survey found that 60% of people watch TV news, 40% read newspaper news, and 25% do both. What is the probability that a randomly selected person watches TV news OR reads newspaper news? | ANSWER: P(TV) = 0.60, P(Newspaper) = 0.40, P(Both) = 0.25. P(TV or Newspaper) = 0.60 + 0.40 - 0.25 = 1.00 - 0.25 = 0.75.
QUESTION: In a batch of 100 mobile phones, 10 have a camera defect (C), 8 have a speaker defect (S), and 3 have both defects. If you pick one phone at random, what is the probability that it has AT LEAST one defect? | ANSWER: P(C) = 10/100 = 0.10, P(S) = 8/100 = 0.08, P(C and S) = 3/100 = 0.03. P(C or S) = P(C) + P(S) - P(C and S) = 0.10 + 0.08 - 0.03 = 0.18 - 0.03 = 0.15.
MCQ
Quick Quiz
If P(A) = 0.5, P(B) = 0.3, and P(A and B) = 0.2, what is P(A or B)?
0.8
0.6
0.7
0.9
The Correct Answer Is:
B
Using the formula P(A or B) = P(A) + P(B) - P(A and B), we get 0.5 + 0.3 - 0.2 = 0.8 - 0.2 = 0.6. Option B is the correct answer.
Real World Connection
In the Real World
Imagine a meteorologist in India predicting rainfall. They might know the probability of a monsoon system forming (Event A) and the probability of local weather conditions causing rain (Event B). Since both can happen, they use this theorem to find the probability of getting rain from either source, helping farmers plan their crops or disaster management teams prepare for floods.
Key Vocabulary
Key Terms
PROBABILITY: The chance of an event happening, usually expressed as a fraction or decimal between 0 and 1. | EVENT: A specific outcome or a set of outcomes in an experiment. | NON-MUTUALLY EXCLUSIVE EVENTS: Events that can happen at the same time or have common outcomes. | OVERLAP: The part where two or more events occur together.
What's Next
What to Learn Next
Great job understanding this! Next, you can explore Conditional Probability, which deals with how the probability of an event changes if another event has already happened. It builds directly on understanding how events interact.
