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What is Complementary Events?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Complementary events are two outcomes of an experiment that are the only two possible outcomes. If one event happens, the other cannot, and together they cover all possibilities. For example, if you flip a coin, getting 'heads' and 'not heads' (which is 'tails') are complementary events.
Simple Example
Quick Example
Imagine you are watching an IPL cricket match. Either your favourite team 'wins' the match, or it 'does not win' (meaning it loses or draws). These two outcomes – your team winning and your team not winning – are complementary events because one or the other must happen, and they are the only two possibilities.
Worked Example
Step-by-Step
Let's say you have a bag with 10 balls: 3 red balls and 7 blue balls. What is the probability of picking a red ball, and what is the probability of not picking a red ball?
---Step 1: Identify the total number of outcomes.
Total balls = 3 (red) + 7 (blue) = 10 balls.
---Step 2: Find the number of favourable outcomes for picking a red ball.
Number of red balls = 3.
---Step 3: Calculate the probability of picking a red ball.
P(Red) = (Number of red balls) / (Total balls) = 3/10.
---Step 4: Find the number of favourable outcomes for not picking a red ball.
Not picking a red ball means picking a blue ball. Number of blue balls = 7.
---Step 5: Calculate the probability of not picking a red ball.
P(Not Red) = (Number of blue balls) / (Total balls) = 7/10.
---Step 6: Check if they are complementary.
P(Red) + P(Not Red) = 3/10 + 7/10 = 10/10 = 1. Yes, they are complementary.
Answer: The probability of picking a red ball is 3/10, and the probability of not picking a red ball is 7/10.
Why It Matters
Understanding complementary events is crucial in many fields like data science, AI, and even economics. For instance, a data scientist might use this to predict if a customer will 'buy a product' or 'not buy a product'. It helps engineers design reliable systems by considering all possible outcomes.
Common Mistakes
MISTAKE: Thinking any two events are complementary. | CORRECTION: Complementary events must be the *only* two possible outcomes and must cover *all* possibilities.
MISTAKE: Forgetting that the sum of probabilities of complementary events is always 1. | CORRECTION: Always remember that P(A) + P(A') = 1, where A' is the complementary event of A.
MISTAKE: Confusing mutually exclusive events with complementary events. | CORRECTION: While complementary events are always mutually exclusive, mutually exclusive events are not always complementary. Complementary events must also cover all possible outcomes.
Practice Questions
Try It Yourself
QUESTION: The probability of rain tomorrow is 0.45. What is the probability that it will NOT rain tomorrow? | ANSWER: P(Not Rain) = 1 - P(Rain) = 1 - 0.45 = 0.55
QUESTION: A standard dice is rolled. What is the probability of rolling a '4'? What is the probability of 'not rolling a 4'? Are these complementary events? | ANSWER: P(rolling a 4) = 1/6. P(not rolling a 4) = 5/6. Yes, they are complementary events because 1/6 + 5/6 = 1.
QUESTION: In a class of 40 students, 25 students like samosas. What is the probability that a randomly chosen student does NOT like samosas? | ANSWER: P(likes samosas) = 25/40 = 5/8. P(does NOT like samosas) = 1 - 5/8 = 3/8. Alternatively, (40-25)/40 = 15/40 = 3/8.
MCQ
Quick Quiz
If the probability of getting a 'Heads' when flipping a coin is 0.5, what is the probability of getting a 'Tails'?
0
0.25
0.5
1
The Correct Answer Is:
C
Getting 'Heads' and 'Tails' are complementary events. The sum of their probabilities must be 1. So, 1 - 0.5 = 0.5.
Real World Connection
In the Real World
When you book a train ticket online via IRCTC, the system might calculate the probability of a 'seat being confirmed' versus 'seat remaining in waiting list'. These are complementary outcomes that help passengers decide if they should book or look for alternatives. Similarly, weather apps predict 'rain' or 'no rain' using this concept.
Key Vocabulary
Key Terms
PROBABILITY: The chance of an event happening. | EVENT: A specific outcome of an experiment. | OUTCOME: A result of an experiment. | MUTUALLY EXCLUSIVE: Events that cannot happen at the same time.
What's Next
What to Learn Next
Great job learning about complementary events! Next, you should explore 'Mutually Exclusive Events' and 'Independent Events'. Understanding these will help you solve even more complex probability problems and build a strong foundation for higher classes.
