What is the Law of Total Probability?

S3-SA3-0445

Grade Level:

Class 9

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

Definition

What is it?

The Law of Total Probability helps us find the overall probability of an event happening when we know how that event depends on several other possible situations. It basically adds up the probabilities of an event occurring under different conditions.

Simple Example

Quick Example

Imagine you want to know the probability of scoring good marks in your Maths exam. This depends on whether you studied hard, studied moderately, or didn't study much. The Law of Total Probability helps combine these different scenarios to find your overall chance of scoring good marks.

Worked Example

Step-by-Step

Let's say a school has two sections for Class 9: Section A and Section B.
Section A has 60 students, and Section B has 40 students.
In Section A, 70% of students passed the Maths test.
In Section B, 80% of students passed the Maths test.
What is the probability that a randomly chosen student from Class 9 passed the Maths test?

Step 1: Find the probability of a student being from Section A.
P(Section A) = Number of students in Section A / Total students = 60 / (60 + 40) = 60 / 100 = 0.6
---
Step 2: Find the probability of a student being from Section B.
P(Section B) = Number of students in Section B / Total students = 40 / 100 = 0.4
---
Step 3: Find the probability of passing the test GIVEN the student is from Section A.
P(Passed | Section A) = 70% = 0.7
---
Step 4: Find the probability of passing the test GIVEN the student is from Section B.
P(Passed | Section B) = 80% = 0.8
---
Step 5: Apply the Law of Total Probability:
P(Passed) = P(Passed | Section A) * P(Section A) + P(Passed | Section B) * P(Section B)
---
Step 6: Calculate the values.
P(Passed) = (0.7 * 0.6) + (0.8 * 0.4)
---
Step 7: Perform the multiplication and addition.
P(Passed) = 0.42 + 0.32 = 0.74

Answer: The probability that a randomly chosen student passed the Maths test is 0.74 or 74%.

Why It Matters

This law is super important for making smart decisions in various fields. Data scientists use it to predict outcomes, engineers use it to assess system reliability, and financial analysts use it to evaluate risks. Learning this helps you think like a problem-solver in careers like AI developer or a market researcher.

Common Mistakes

MISTAKE: Confusing P(A and B) with P(A | B). Students might multiply P(A) * P(B) directly. | CORRECTION: Remember P(A | B) is the probability of A happening GIVEN B has already happened. The Law of Total Probability uses conditional probabilities carefully.

MISTAKE: Forgetting that the 'conditions' or 'partitions' must cover all possibilities and be mutually exclusive. | CORRECTION: Ensure that the events you're conditioning on (like Section A and Section B) cover all cases (all students are in either A or B) and don't overlap (no student is in both A and B).

MISTAKE: Incorrectly adding probabilities without considering the 'weight' or initial probability of each condition. | CORRECTION: Always multiply the conditional probability P(Event | Condition) by the probability of that condition P(Condition) before summing them up.

Practice Questions

Try It Yourself

QUESTION: A factory produces LED bulbs using two machines, M1 and M2. M1 produces 60% of the bulbs, and M2 produces 40%. 2% of bulbs from M1 are defective, and 3% of bulbs from M2 are defective. What is the probability that a randomly chosen bulb is defective? | ANSWER: P(Defective) = (0.02 * 0.60) + (0.03 * 0.40) = 0.012 + 0.012 = 0.024 or 2.4%

QUESTION: In a game, you can choose one of two bags. Bag 1 has 5 red balls and 5 blue balls. Bag 2 has 3 red balls and 7 blue balls. You pick Bag 1 with a probability of 0.6 and Bag 2 with a probability of 0.4. What is the probability that you pick a red ball? | ANSWER: P(Red) = P(Red | Bag 1) * P(Bag 1) + P(Red | Bag 2) * P(Bag 2) = (5/10 * 0.6) + (3/10 * 0.4) = (0.5 * 0.6) + (0.3 * 0.4) = 0.30 + 0.12 = 0.42

QUESTION: An online delivery service has three types of vehicles: bikes (60% of deliveries), scooters (30%), and cars (10%). The probability of a late delivery is 5% for bikes, 3% for scooters, and 1% for cars. If a delivery is chosen at random, what is the probability that it will be on time? | ANSWER: P(Late) = (0.05 * 0.60) + (0.03 * 0.30) + (0.01 * 0.10) = 0.03 + 0.009 + 0.001 = 0.04. So, P(On Time) = 1 - P(Late) = 1 - 0.04 = 0.96 or 96%.

MCQ

Quick Quiz

Which of the following best describes the Law of Total Probability?

It finds the probability of two independent events happening together.

It calculates the overall probability of an event by summing probabilities across mutually exclusive and exhaustive conditions.

It determines the probability of an event given another event has already occurred.

It is used to find the probability of the union of two events.

The Correct Answer Is:

Option B correctly states that the Law of Total Probability sums up probabilities of an event under different, non-overlapping conditions that cover all possibilities. Options A, C, and D describe other probability concepts like independence, conditional probability, and union of events.

Real World Connection

In the Real World

Think about predicting the success rate of a new mobile app launch in India. The success might depend on the city (Tier 1, Tier 2, Tier 3), the marketing strategy used in each city, and the local internet penetration. Data analysts use the Law of Total Probability to combine these factors and estimate the overall app success, helping companies like Flipkart or Zomato plan better.

Key Vocabulary

Key Terms

Probability: The chance of an event happening | Conditional Probability: The probability of an event occurring given another event has already occurred | Mutually Exclusive Events: Events that cannot happen at the same time | Exhaustive Events: A set of events that covers all possible outcomes | Partition: A division of a sample space into mutually exclusive and exhaustive events

What's Next

What to Learn Next

Once you've mastered the Law of Total Probability, you're ready to explore Bayes' Theorem! It's a powerful concept that builds on this law, allowing you to update probabilities based on new information, which is super useful in AI and machine learning.