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What is the Probability of Complementary Events?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Complementary events are two events that are the only two possible outcomes for an experiment. If one event happens, the other cannot, and together they cover all possibilities. The sum of their probabilities is always 1.
Simple Example
Quick Example
Imagine your favourite cricket team, India, is playing a match. There are only two main outcomes: India wins, or India does not win (meaning they lose or draw). If the probability of India winning is 0.7, then the probability of India not winning must be 1 - 0.7 = 0.3.
Worked Example
Step-by-Step
Suppose you are checking if your mobile phone battery is charged enough for a full day. The probability that your phone battery will last all day is 0.85. What is the probability that it will NOT last all day?
Step 1: Identify the event and its probability. Event A = 'Phone battery lasts all day'. P(A) = 0.85.
---Step 2: Identify the complementary event. The complementary event, A', is 'Phone battery does NOT last all day'.
---Step 3: Recall the rule for complementary events: P(A) + P(A') = 1.
---Step 4: Substitute the known probability into the formula: 0.85 + P(A') = 1.
---Step 5: Solve for P(A'): P(A') = 1 - 0.85.
---Step 6: Calculate the result: P(A') = 0.15.
Answer: The probability that your phone battery will NOT last all day is 0.15.
Why It Matters
Understanding complementary events is crucial in fields like AI/ML to predict outcomes or in FinTech to assess risks. Engineers use it to calculate the probability of a system failing versus succeeding. Even doctors use it to understand the chances of a treatment working or not working.
Common Mistakes
MISTAKE: Adding the probabilities of two events that are not complementary. For example, 'getting a 6 on a dice' and 'getting an even number' are not complementary. | CORRECTION: Complementary events must be the ONLY two possible outcomes and cannot happen at the same time. Always check if their probabilities sum to 1.
MISTAKE: Forgetting that the sum of probabilities for complementary events is exactly 1. Sometimes students might use 100% instead of 1, or some other number. | CORRECTION: Probabilities are always between 0 and 1 (inclusive). For complementary events A and A', P(A) + P(A') must always equal 1.
MISTAKE: Confusing mutually exclusive events with complementary events. All complementary events are mutually exclusive, but not all mutually exclusive events are complementary. | CORRECTION: Complementary events must also be 'exhaustive', meaning they cover all possible outcomes. Mutually exclusive events just mean they can't happen at the same time.
Practice Questions
Try It Yourself
QUESTION: The probability of rain tomorrow in Bengaluru is 0.6. What is the probability that it will NOT rain tomorrow? | ANSWER: 0.4
QUESTION: A student is picked from a class. The probability that the student is a boy is 0.55. What is the probability that the student is a girl? | ANSWER: 0.45
QUESTION: In a game, the probability of winning a prize is 1/4. What is the probability of NOT winning a prize? Express your answer as a decimal. | ANSWER: 0.75
MCQ
Quick Quiz
If the probability of a new electric car model succeeding in the market is 0.72, what is the probability of it NOT succeeding?
0.28
0.72
0.18
1
The Correct Answer Is:
A
For complementary events, the sum of their probabilities is 1. So, P(not succeeding) = 1 - P(succeeding) = 1 - 0.72 = 0.28.
Real World Connection
In the Real World
Imagine a weather app like AccuWeather or Google Weather on your phone. When it shows '40% chance of rain', it automatically implies there's a 60% chance of 'no rain'. This uses the concept of complementary events to give you a complete picture of the weather possibility, helping you decide whether to carry an umbrella or not.
Key Vocabulary
Key Terms
EVENT: A specific outcome of an experiment. | PROBABILITY: The chance of an event happening, expressed as a number between 0 and 1. | COMPLEMENTARY EVENTS: Two events that are the only possible outcomes and whose probabilities sum to 1. | OUTCOME: A single possible result of an experiment.
What's Next
What to Learn Next
Great job understanding complementary events! Next, you should explore 'Mutually Exclusive Events' and 'Independent Events'. These concepts will help you understand how different events interact and are crucial for solving more complex probability problems.
