What is Multiplication Rule for Independent Events? | Class 12

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What is the Multiplication Rule for Independent Events?

Grade Level:

Class 12

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Definition

What is it?

The Multiplication Rule for Independent Events helps us find the probability that two or more independent events will *all* happen. Independent events are those where the outcome of one event does not affect the outcome of another event. You simply multiply their individual probabilities.

Simple Example

Quick Example

Imagine you toss a fair coin and roll a standard six-sided dice. Getting a 'Head' on the coin and rolling a '4' on the dice are independent events. The probability of getting a Head is 1/2, and the probability of rolling a 4 is 1/6. To find the probability of both happening, you multiply (1/2) * (1/6) = 1/12.

Worked Example

Step-by-Step

Let's say a famous Indian cricketer, Virat Kohli, has a 0.8 probability of hitting a boundary in an over, and Rohit Sharma has a 0.7 probability of hitting a boundary in a *different* over. What is the probability that both will hit a boundary in their respective overs?

Step 1: Identify the events and their probabilities.
Event A: Virat hits a boundary. P(A) = 0.8
Event B: Rohit hits a boundary. P(B) = 0.7

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Step 2: Confirm if the events are independent. Since they are hitting boundaries in different overs, one's performance doesn't affect the other's. So, they are independent.

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Step 3: Apply the Multiplication Rule for Independent Events: P(A and B) = P(A) * P(B).

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Step 4: Substitute the probabilities and calculate.
P(Virat hits boundary AND Rohit hits boundary) = 0.8 * 0.7

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Step 5: Calculate the final probability.
0.8 * 0.7 = 0.56

Answer: The probability that both Virat Kohli and Rohit Sharma will hit a boundary in their respective overs is 0.56.

Why It Matters

This rule is super important in many fields, from predicting weather patterns to designing safe airplanes. Engineers use it to calculate the chance of multiple parts failing, while doctors use it to understand disease risks. Knowing this helps you make smarter decisions and build reliable systems.

Common Mistakes

MISTAKE: Adding probabilities instead of multiplying them. | CORRECTION: Remember, 'AND' usually means multiply for independent events, while 'OR' often means add (for mutually exclusive events).

MISTAKE: Applying the rule to dependent events. | CORRECTION: Always check if events are truly independent. If one event affects the other, you need a different rule (conditional probability).

MISTAKE: Not converting percentages to decimals or fractions before multiplying. | CORRECTION: Always express probabilities as decimals (e.g., 70% as 0.7) or fractions (e.g., 1/2) before performing calculations.

Practice Questions

Try It Yourself

QUESTION: You have a bag with 5 red marbles and 5 blue marbles. You pick one marble, put it back, and then pick another. What is the probability of picking a red marble both times? | ANSWER: 1/4 or 0.25

QUESTION: A traffic light at one junction is green 60% of the time. A traffic light at a different, unrelated junction is green 75% of the time. What is the probability that both lights are green when you approach them? | ANSWER: 0.45 or 45%

QUESTION: A robot in a factory has a 0.95 chance of successfully assembling a part. If it tries to assemble three different parts, and each attempt is independent, what is the probability that all three attempts are successful? | ANSWER: 0.95 * 0.95 * 0.95 = 0.857375

MCQ

Quick Quiz

What is the probability of flipping a coin and getting a 'Head' AND rolling a fair six-sided die and getting an even number?

2026-01-02T00:00:00.000Z

2026-01-04T00:00:00.000Z

2026-01-06T00:00:00.000Z

2026-01-12T00:00:00.000Z

The Correct Answer Is:

The probability of getting a Head is 1/2. The probability of getting an even number (2, 4, or 6) on a die is 3/6 = 1/2. Since the events are independent, we multiply (1/2) * (1/2) = 1/4.

Real World Connection

In the Real World

In climate science, scientists use this rule to predict the combined probability of independent weather events, like a certain temperature range AND a certain rainfall amount, to model future climate scenarios. This helps predict monsoon patterns or extreme weather events, which are crucial for Indian agriculture and disaster preparedness.

Key Vocabulary

Key Terms

Probability: The likelihood of an event happening | Independent Events: Events where the outcome of one does not affect the other | Dependent Events: Events where the outcome of one influences the other | Outcome: A possible result of an experiment or event

What's Next

What to Learn Next

Great job learning about independent events! Next, you should explore the 'Multiplication Rule for Dependent Events'. This will teach you how to calculate probabilities when one event's outcome changes the chances of another, which is a common scenario in real life.