What is the Addition Theorem of Probability?

S7-SA3-0315

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 helps us find the chance of at least one of two or more events happening. It tells us how to combine probabilities when events might overlap or be separate.

Simple Example

Quick Example

Imagine you want to know the chance of getting an even number OR a number greater than 4 when you roll a single dice. An even number can be 2, 4, 6. A number greater than 4 can be 5, 6. Notice that 6 is in both lists! The Addition Theorem helps us count 6 only once.

Worked Example

Step-by-Step

Let's find the probability of drawing a King OR a Red card from a standard deck of 52 cards.

Step 1: Find the probability of drawing a King (Event A).
There are 4 Kings in a deck of 52 cards. So, P(A) = 4/52.

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Step 2: Find the probability of drawing a Red card (Event B).
There are 26 Red cards (13 Hearts + 13 Diamonds) in a deck of 52 cards. So, P(B) = 26/52.

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Step 3: Find the probability of drawing a card that is BOTH a King AND a Red card (Event A and B).
There are 2 Red Kings (King of Hearts, King of Diamonds) in a deck. So, P(A and B) = 2/52.

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Step 4: Apply the Addition Theorem formula: P(A or B) = P(A) + P(B) - P(A and B).
Substitute the values: P(King or Red) = 4/52 + 26/52 - 2/52.

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Step 5: Calculate the result.
P(King or Red) = (4 + 26 - 2) / 52 = 28/52.

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Step 6: Simplify the fraction.
28/52 can be simplified by dividing both by 4, which gives 7/13.

Answer: The probability of drawing a King or a Red card is 7/13.

Why It Matters

This theorem is super important for making smart decisions in fields like AI, FinTech, and even Medicine. For example, doctors use it to understand the chances of a patient having one disease OR another based on symptoms. Engineers use it to calculate the reliability of systems where different parts might fail.

Common Mistakes

MISTAKE: Simply adding probabilities of two events without considering overlap. | CORRECTION: Always subtract the probability of both events happening together if they are not mutually exclusive.

MISTAKE: Confusing 'OR' with 'AND'. | CORRECTION: The Addition Theorem is for 'OR' (at least one event). For 'AND' (both events), you usually multiply probabilities.

MISTAKE: Applying the simplified P(A or B) = P(A) + P(B) when events are NOT mutually exclusive. | CORRECTION: Remember the full formula P(A or B) = P(A) + P(B) - P(A and B). The simplified version only works if P(A and B) is 0 (mutually exclusive events).

Practice Questions

Try It Yourself

QUESTION: What is the probability of drawing a Heart OR a Spade from a standard deck of 52 cards? | ANSWER: 26/52 or 1/2

QUESTION: In a class, 20% of students like Cricket, 30% like Football, and 10% like both. What is the probability that a randomly chosen student likes Cricket OR Football? | ANSWER: 0.4 or 40%

QUESTION: A bag contains 10 balls: 4 red, 3 blue, and 3 green. What is the probability of picking a red ball OR a blue ball? | ANSWER: 7/10

MCQ

Quick Quiz

If P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, what is P(A or B)?

0.7

0.9

0.3

1.1

The Correct Answer Is:

Using the Addition Theorem, P(A or B) = P(A) + P(B) - P(A and B) = 0.5 + 0.4 - 0.2 = 0.9 - 0.2 = 0.7. Options B, C, and D are incorrect calculations.

Real World Connection

In the Real World

Imagine you're an analyst for a weather forecasting app like Mausam. You might use the Addition Theorem to predict the probability of 'rain OR strong winds' in your city, Chennai, tomorrow. This helps people plan their day, whether to carry an umbrella or secure their balcony plants. It's also used in cricket analytics to calculate the chance of a batsman scoring a fifty OR hitting a six in the next few overs.

Key Vocabulary

Key Terms

PROBABILITY: The chance of an event happening, expressed as a number between 0 and 1. | EVENT: A specific outcome or set of outcomes in an experiment. | MUTUALLY EXCLUSIVE EVENTS: Events that cannot happen at the same time (e.g., getting a head and a tail on a single coin toss). | OVERLAPPING EVENTS: Events that can happen at the same time (e.g., drawing a King and a Red card).

What's Next

What to Learn Next

Next, you can explore Conditional Probability. This concept builds on the Addition Theorem by looking at how the probability of an event changes if we already know another event has occurred, which is super useful for real-world predictions!