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What is the Complement of an Event?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The complement of an event is everything that can happen EXCEPT for that event itself. If an event 'A' happens, its complement 'A'' (read as 'A-prime' or 'A-complement') means 'A does not happen'. Together, an event and its complement cover all possible outcomes.
Simple Example
Quick Example
Imagine you are watching an IPL cricket match. Let Event A be 'Chennai Super Kings win the match'. Then, the complement of Event A (A') would be 'Chennai Super Kings do NOT win the match', which means they either lose or draw. These are the only two possibilities for CSK in that match.
Worked Example
Step-by-Step
Problem: A bag contains 5 red balls and 3 blue balls. If you pick one ball randomly, what is the probability of picking a blue ball, and what is the probability of its complement?
Step 1: Find the total number of balls.
Total balls = 5 (red) + 3 (blue) = 8 balls.
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Step 2: Define Event B as 'picking a blue ball'.
Number of blue balls = 3.
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Step 3: Calculate the probability of Event B.
P(B) = (Number of blue balls) / (Total number of balls) = 3/8.
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Step 4: Define the complement of Event B, which is B' (not picking a blue ball).
This means picking any ball that is NOT blue. In this case, it means picking a red ball.
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Step 5: Calculate the probability of B' (not picking a blue ball).
P(B') = (Number of red balls) / (Total number of balls) = 5/8.
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Step 6: Verify that P(B) + P(B') = 1.
P(B) + P(B') = 3/8 + 5/8 = 8/8 = 1.
Answer: The probability of picking a blue ball is 3/8, and the probability of its complement (not picking a blue ball) is 5/8.
Why It Matters
Understanding complements helps us calculate probabilities of 'not happening' which is crucial in AI/ML for error analysis, in Data Science for filtering out unwanted data, and in Engineering for reliability checks. For example, engineers use it to predict the probability of a machine NOT failing, which is vital for safety.
Common Mistakes
MISTAKE: Thinking the complement is just the opposite outcome (e.g., win vs. lose) when there might be more possibilities (e.g., win, lose, draw). | CORRECTION: The complement includes ALL other possible outcomes that are not the event itself.
MISTAKE: Adding up the probabilities of an event and its complement and getting a number other than 1. | CORRECTION: The sum of the probability of an event and the probability of its complement MUST always be exactly 1 (or 100%).
MISTAKE: Confusing 'complement' with 'mutually exclusive events'. | CORRECTION: While an event and its complement are mutually exclusive (they can't happen at the same time), they also cover ALL possible outcomes together. Mutually exclusive events don't always cover all outcomes.
Practice Questions
Try It Yourself
QUESTION: If the probability of rain tomorrow is 0.4, what is the probability that it will NOT rain tomorrow? | ANSWER: 0.6
QUESTION: A spinner has 8 equal sections numbered 1 to 8. What is the probability of spinning an even number? What is the probability of its complement? | ANSWER: P(Even) = 4/8 = 1/2. P(Not Even) = 4/8 = 1/2.
QUESTION: In a class of 40 students, 15 students got A grade in Maths. What is the probability that a randomly chosen student did NOT get an A grade in Maths? | ANSWER: P(A grade) = 15/40. P(Not A grade) = 1 - 15/40 = 25/40 = 5/8.
MCQ
Quick Quiz
If the probability of a new mobile phone having a manufacturing defect is 0.05, what is the probability that a new mobile phone does NOT have a manufacturing defect?
0.05
0.95
0.5
1
The Correct Answer Is:
B
The probability of an event and its complement always sum to 1. So, if P(defect) = 0.05, then P(no defect) = 1 - 0.05 = 0.95. Options A, C, and D are incorrect calculations.
Real World Connection
In the Real World
When you book a train ticket on IRCTC, the system might calculate the probability of your ticket getting confirmed. The complement would be the probability of your ticket NOT getting confirmed (remaining in waitlist or getting cancelled). This helps you decide if you should look for other travel options.
Key Vocabulary
Key Terms
EVENT: A specific outcome or set of outcomes in an experiment | PROBABILITY: A measure of how likely an event is to occur, between 0 and 1 | OUTCOME: A possible result of an experiment | COMPLEMENT: All outcomes that are not part of a specific event
What's Next
What to Learn Next
Great job understanding complements! Next, you can explore 'Mutually Exclusive Events' and 'Independent Events'. These concepts build on the idea of different outcomes and how their probabilities combine or don't combine, which is super useful for solving more complex probability problems.
