What is Multiplication Theorem of Probability? | Simple Explanation for Class 8

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What is the Multiplication Theorem of Probability for Dependent Events?

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Multiplication Theorem of Probability for Dependent Events helps us find the chance of two events happening one after the other, where the first event affects the second. It means the outcome of the first event changes the possibilities for the second event. We calculate the probability of the first event, then multiply it by the probability of the second event happening GIVEN that the first event has already occurred.

Simple Example

Quick Example

Imagine you have a bag with 5 red ladoos and 5 green ladoos. If you pick one red ladoo and eat it (Event A), the total number of ladoos in the bag changes. Now, if you want to pick another red ladoo (Event B), the chance of picking it is different because there are fewer red ladoos and fewer total ladoos left. This is a dependent event.

Worked Example

Step-by-Step

Let's say a cricket team has a box with 10 cricket balls: 7 new balls and 3 old balls. The bowler picks one ball, bowls with it, and does NOT put it back. What is the probability that the bowler picks two new balls in a row?

Step 1: Find the probability of picking the first new ball.
Total balls = 10. New balls = 7.
P(First New Ball) = 7/10
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Step 2: After picking one new ball, there are now 9 balls left in the box. And there are only 6 new balls left.
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Step 3: Find the probability of picking a second new ball, GIVEN that the first one was new and not replaced.
P(Second New Ball | First was New) = 6/9
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Step 4: Multiply the probabilities from Step 1 and Step 3.
P(First New AND Second New) = P(First New) * P(Second New | First was New)
= (7/10) * (6/9)
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Step 5: Calculate the product.
= 42/90
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Step 6: Simplify the fraction.
= 7/15

Answer: The probability of picking two new balls in a row is 7/15.

Why It Matters

Understanding dependent probability helps scientists predict outcomes in AI/ML models, like predicting customer behavior based on past choices. Data scientists use it to analyze patterns in large datasets, helping companies make smart decisions. It's also vital in fields like cryptography to secure online transactions and in engineering to assess system reliability.

Common Mistakes

MISTAKE: Treating dependent events as independent and simply multiplying their individual probabilities without considering the change. | CORRECTION: Always check if the first event affects the conditions for the second event. If it does, adjust the probability of the second event accordingly.

MISTAKE: Forgetting to reduce the total number of outcomes (denominator) AND the number of favorable outcomes (numerator) for the second event after the first event occurs. | CORRECTION: After the first event, mentally or physically remove the outcome from the sample space before calculating the probability for the second event.

MISTAKE: Confusing 'AND' (multiplication) with 'OR' (addition) in probability questions. | CORRECTION: The Multiplication Theorem is used when you want to find the probability of Event A AND then Event B happening. If it's Event A OR Event B, you'd typically use addition (for mutually exclusive events).

Practice Questions

Try It Yourself

QUESTION: A box contains 4 red pens and 6 blue pens. If you pick a pen, keep it, and then pick another pen, what is the probability that both pens picked are red? | ANSWER: 4/10 * 3/9 = 12/90 = 2/15

QUESTION: In a class of 20 students, 12 like Maths and 8 like Science. If two students are randomly chosen one after another without replacement, what is the probability that the first student likes Maths and the second student likes Science? | ANSWER: 12/20 * 8/19 = 96/380 = 24/95

QUESTION: A basket has 8 apples and 7 oranges. You pick one fruit, eat it, then pick a second fruit, eat it, and then pick a third fruit. What is the probability that you pick an apple, then an orange, then another apple? | ANSWER: 8/15 * 7/14 * 7/13 = 392/2730 = 28/195

MCQ

Quick Quiz

You have a deck of 52 playing cards. What is the probability of drawing a King, then drawing another King without replacing the first one?

4/52 * 4/52

4/52 + 3/51

4/52 * 3/51

4/52 + 4/52

The Correct Answer Is:

When the first King is drawn and not replaced, there are only 3 Kings left and 51 total cards. So, the probability of the second King depends on the first draw. We multiply the probability of the first King (4/52) by the probability of the second King given the first (3/51).

Real World Connection

In the Real World

Imagine a food delivery app like Zomato or Swiggy predicting if a customer will order a dessert after ordering a main course. If a customer orders a pizza (Event A), the probability of them ordering a chocolate lava cake (Event B) might increase. Data scientists use dependent probability to build these recommendation systems, making your app experience better and helping businesses grow.

Key Vocabulary

Key Terms

Dependent Events: Events where the outcome of one event affects the outcome of another. | Probability: The chance of an event happening, expressed as a fraction or decimal. | Without Replacement: When an item is selected and not put back, changing the total possibilities. | Conditional Probability: The probability of an event happening, given that another event has already occurred.

What's Next

What to Learn Next

Great job understanding dependent events! Next, you can explore Bayes' Theorem, which takes this idea even further to update probabilities based on new evidence. This will open doors to understanding how AI systems learn and make predictions.