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What is Complementary Event?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
A complementary event is everything that can happen in an experiment EXCEPT for a specific event. If an event 'A' occurs, its complementary event 'A'' (read as 'A prime' or 'A complement') includes all outcomes where 'A' does not occur. Together, an event and its complement cover all possible outcomes of an experiment.
Simple Example
Quick Example
Imagine you are watching an IPL cricket match. Let 'Event A' be that 'Chennai Super Kings (CSK) wins the match'. The complementary event, 'A'', would be 'CSK does NOT win the match'. This means CSK either loses or the match ends in a tie. So, 'CSK wins' and 'CSK does not win' are complementary events.
Worked Example
Step-by-Step
PROBLEM: You have a bag with 10 laddoos: 3 besan laddoos and 7 motichoor laddoos. If you pick one laddoo randomly, what is the probability of NOT picking a besan laddoo?
Step 1: Define the event. Let Event B be 'picking a besan laddoo'.
---Step 2: Calculate the total number of possible outcomes. There are 10 laddoos in total.
---Step 3: Calculate the number of favourable outcomes for Event B. There are 3 besan laddoos.
---Step 4: Calculate the probability of Event B: P(B) = (Number of besan laddoos) / (Total laddoos) = 3/10.
---Step 5: Define the complementary event. The complementary event B' is 'NOT picking a besan laddoo'.
---Step 6: Use the formula: P(B') = 1 - P(B).
---Step 7: Substitute the value: P(B') = 1 - 3/10 = 10/10 - 3/10 = 7/10.
---Answer: The probability of NOT picking a besan laddoo is 7/10.
Why It Matters
Understanding complementary events is crucial in many fields. For example, in AI/ML, it helps in calculating the probability of an error not happening. In medicine, doctors use it to find the chance a patient does NOT have a certain disease. Knowing this concept can even help engineers design safer systems by calculating the probability of a system NOT failing.
Common Mistakes
MISTAKE: Thinking an event and its opposite are always complementary. | CORRECTION: An event and its complement must cover ALL possible outcomes and have no outcomes in common. For example, 'getting a 6' and 'getting a 4' on a dice are opposites, but not complementary, because 'getting a 1, 2, 3, or 5' are also possible outcomes.
MISTAKE: Adding probabilities of an event and its complement to get a value other than 1. | CORRECTION: The sum of the probability of an event and the probability of its complementary event must always be exactly 1. P(A) + P(A') = 1.
MISTAKE: Confusing mutually exclusive events with complementary events. | CORRECTION: While complementary events are always mutually exclusive (they can't happen at the same time), not all mutually exclusive events are complementary. Complementary events must also be 'exhaustive', meaning they cover all possibilities together.
Practice Questions
Try It Yourself
QUESTION: You are playing Ludo. What is the probability of NOT rolling a '1' on a single die? | ANSWER: 5/6
QUESTION: A traffic light at a busy intersection in Mumbai is green for 40 seconds, yellow for 5 seconds, and red for 15 seconds in a 60-second cycle. What is the probability that the light is NOT green when you approach it? | ANSWER: 20/60 or 1/3
QUESTION: In a class of 50 students, 30 students like samosas, 15 students like jalebis, and 5 students like both. If a student is chosen randomly, what is the probability that the student likes NEITHER samosas NOR jalebis? (Hint: First find the probability that a student likes at least one of them). | ANSWER: 10/50 or 1/5
MCQ
Quick Quiz
If the probability of rain in Bengaluru tomorrow is 0.35, what is the probability that it will NOT rain?
0.35
0.65
1
0.7
The Correct Answer Is:
B
The probability of an event and its complement always sum to 1. So, P(not rain) = 1 - P(rain) = 1 - 0.35 = 0.65. Option A is the probability of rain, not its complement.
Real World Connection
In the Real World
Imagine a weather app on your phone that shows the chance of rain. If it says there's a 30% chance of rain, it's indirectly telling you there's a 70% chance it will NOT rain. This uses complementary events. Similarly, when ISRO launches a rocket, engineers calculate the probability of a successful launch, and also the complementary probability of an unsuccessful launch to plan for all scenarios.
Key Vocabulary
Key Terms
Probability: The chance of an event happening. | Event: A specific outcome or set of outcomes in an experiment. | Outcome: A possible result of an experiment. | Sample Space: The set of all possible outcomes 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 'Conditional Probability'. This concept builds on what you've learned by showing how the probability of one event changes if another event has already happened. It's super useful for understanding real-world situations!
