What is Complementary Probability?

S7-SA3-0417

Grade Level:

Class 12

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

Definition

What is it?

Complementary probability helps us find the chance of an event NOT happening if we know the chance of it happening. It's based on the idea that an event either happens or it doesn't, and the total probability of all possibilities is always 1.

Simple Example

Quick Example

Imagine you are waiting for a bus. The probability that the bus arrives on time is 0.8 (or 80%). The complementary probability is the chance that the bus does NOT arrive on time. Since it either arrives on time or it doesn't, this would be 1 - 0.8 = 0.2 (or 20%).

Worked Example

Step-by-Step

PROBLEM: In a cricket match, the probability of Team India winning is 0.65. What is the probability that Team India does NOT win?
---STEP 1: Identify the event and its given probability. The event is 'Team India wins', and its probability P(Win) = 0.65.
---STEP 2: Understand that 'Team India does NOT win' is the complement of the event 'Team India wins'.
---STEP 3: Use the complementary probability formula: P(not A) = 1 - P(A).
---STEP 4: Substitute the given probability into the formula. P(not Win) = 1 - P(Win) = 1 - 0.65.
---STEP 5: Calculate the result. 1 - 0.65 = 0.35.
---ANSWER: The probability that Team India does NOT win is 0.35.

Why It Matters

Understanding complementary probability is key in fields like AI/ML to predict failures, in Medicine to calculate risks of a disease not occurring, and in FinTech to assess the chance of an investment not losing money. Engineers use it to ensure systems don't fail, and climate scientists predict events not happening. It helps make better decisions in many real-world careers.

Common Mistakes

MISTAKE: Adding the probability of an event to 1 instead of subtracting. For example, calculating P(not A) = 1 + P(A). | CORRECTION: Always subtract the probability of the event from 1 to find its complement: P(not A) = 1 - P(A).

MISTAKE: Confusing mutually exclusive events with complementary events. Thinking any two events that can't happen at the same time are complementary. | CORRECTION: Complementary events MUST cover ALL possibilities. Event A and its complement 'not A' are the ONLY two outcomes. Their probabilities must add up to 1.

MISTAKE: Using percentages directly without converting to decimals. For example, 1 - 65% = 35% (which is correct numerically but often misunderstood as 1 - 0.65 = 0.35). | CORRECTION: Always convert percentages to decimals (divide by 100) before using them in probability calculations to avoid errors and ensure consistency. So, 65% becomes 0.65.

Practice Questions

Try It Yourself

QUESTION: The probability of a new mobile phone model being successful in the Indian market is 0.7. What is the probability that it will NOT be successful? | ANSWER: 0.3

QUESTION: A bag contains only red and blue marbles. The probability of picking a red marble is 3/7. What is the probability of picking a blue marble? | ANSWER: 4/7

QUESTION: A student has a 20% chance of getting an 'A' grade in Math and a 30% chance of getting a 'B' grade. Assuming only A, B, C, D, F grades are possible and these are the only two events given, what is the probability of NOT getting an 'A' or 'B' grade? | ANSWER: 0.5 (or 50%)

MCQ

Quick Quiz

If the probability of rain tomorrow is 0.45, what is the probability that it will NOT rain tomorrow?

0.45

0.55

The Correct Answer Is:

The probability of an event and its complement must add up to 1. So, P(not rain) = 1 - P(rain) = 1 - 0.45 = 0.55.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola/Uber, it estimates your arrival time. This estimate has a probability of being correct. Complementary probability helps the app understand the chance of you NOT arriving on time, perhaps due to traffic. This helps in real-time re-routing or informing the driver/passenger, making your daily commute smoother.

Key Vocabulary

Key Terms

COMPLEMENTARY EVENT: The event that an original event does NOT happen. | PROBABILITY: The measure of how likely an event is to occur. | TOTAL PROBABILITY: The sum of probabilities of all possible outcomes, which is always 1. | OUTCOME: A possible result of an experiment or event.

What's Next

What to Learn Next

Next, you can explore 'Mutually Exclusive Events' and 'Independent Events'. These concepts build on understanding basic probability and complements, helping you solve more complex problems where multiple events happen together or separately.