What is Total Probability Theorem? | Simple Explanation for Class 12

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What is the Theorem of Total Probability?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The Theorem of Total Probability helps us find the overall probability of an event happening when there are several different ways it can occur. It works by combining the probabilities of that event happening under each of those different conditions.

Simple Example

Quick Example

Imagine you want to know the chance of a student passing their Class 12 board exam. Some students prepare by going to coaching classes, and others study on their own. The Theorem of Total Probability helps calculate the overall passing chance by considering both groups and their individual passing probabilities.

Worked Example

Step-by-Step

PROBLEM: A factory makes LED bulbs. Machine A produces 60% of the bulbs, and Machine B produces 40%. 2% of bulbs from Machine A are defective, and 3% of bulbs from Machine B are defective. What is the total probability that a randomly chosen bulb is defective?

STEP 1: Identify the events. Let 'D' be the event that a bulb is defective. Let 'A' be the event that a bulb is from Machine A, and 'B' be the event that a bulb is from Machine B.
---STEP 2: Write down the given probabilities.
P(A) = 0.60 (Machine A produces 60%)
P(B) = 0.40 (Machine B produces 40%)
P(D|A) = 0.02 (2% defective from Machine A)
P(D|B) = 0.03 (3% defective from Machine B)
---STEP 3: Apply the formula for Total Probability: P(D) = P(D|A) * P(A) + P(D|B) * P(B).
---STEP 4: Substitute the values into the formula.
P(D) = (0.02 * 0.60) + (0.03 * 0.40)
---STEP 5: Calculate the products.
0.02 * 0.60 = 0.012
0.03 * 0.40 = 0.012
---STEP 6: Add the results.
P(D) = 0.012 + 0.012 = 0.024
---ANSWER: The total probability that a randomly chosen bulb is defective is 0.024 or 2.4%.

Why It Matters

This theorem is super useful in fields like AI/ML to predict outcomes, in medicine to understand disease risks, and in FinTech to assess investment chances. Engineers use it to calculate system reliability, and even climate scientists predict weather patterns! Learning this can open doors to exciting careers in data science, medical research, or financial analysis.

Common Mistakes

MISTAKE: Confusing P(A|B) with P(B|A). | CORRECTION: Remember P(A|B) means the probability of A happening GIVEN B has already happened, not the other way around. Read the 'given' condition carefully.

MISTAKE: Not ensuring the 'partitions' (like Machine A and Machine B) cover all possibilities and are mutually exclusive. | CORRECTION: The events you are partitioning by (e.g., source of the bulb) must add up to 100% and cannot overlap.

MISTAKE: Forgetting to multiply the conditional probability by the probability of the condition itself. | CORRECTION: The formula is P(Event) = P(Event|Condition1) * P(Condition1) + P(Event|Condition2) * P(Condition2). Don't miss the P(Condition) part.

Practice Questions

Try It Yourself

QUESTION: In a school, 70% of students are boys and 30% are girls. 10% of boys wear glasses, and 20% of girls wear glasses. What is the probability that a randomly chosen student wears glasses? | ANSWER: 0.13 or 13%

QUESTION: A tiffin service delivers meals. 80% of deliveries are by bike, 20% by cycle. 5% of bike deliveries are late, and 15% of cycle deliveries are late. What is the total probability that a random delivery is late? | ANSWER: 0.07 or 7%

QUESTION: A mobile phone company gets its screens from three suppliers: S1, S2, and S3. S1 supplies 50% of screens, S2 supplies 30%, and S3 supplies 20%. The defect rates are 1% for S1, 2% for S2, and 3% for S3. If a phone is assembled, what is the probability that its screen is defective? | ANSWER: 0.017 or 1.7%

MCQ

Quick Quiz

Which of the following scenarios is best solved using the Theorem of Total Probability?

Finding the probability of getting heads twice in a row with a fair coin.

Calculating the overall chance of rain, given it could be a cloudy day or a clear day.

Determining the probability of drawing a red card OR a king from a deck.

Finding the probability that a student passes an exam GIVEN they studied for 10 hours.

The Correct Answer Is:

Option B fits the Total Probability Theorem because it involves finding the overall probability of an event (rain) by considering different, mutually exclusive conditions (cloudy day or clear day) that cover all possibilities. The other options are for independent events, unions of events, or conditional probability directly.

Real World Connection

In the Real World

Imagine an app like 'Swiggy' or 'Zomato' trying to predict if a food order will be delivered on time. They use the Theorem of Total Probability to calculate the overall chance of on-time delivery. They might consider factors like whether the delivery is by scooter or cycle, during peak or off-peak hours, or in heavy traffic, each with its own probability of being on time or late. This helps them give you accurate delivery estimates!

Key Vocabulary

Key Terms

Probability: The chance of an event happening. | Conditional Probability: The probability of an event happening, given that 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.

What's Next

What to Learn Next

Great job understanding Total Probability! Next, you should explore Bayes' Theorem. It builds directly on this concept and helps you update probabilities based on new information, which is super powerful in data science and AI. Keep learning and growing!