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What is Mutually Exclusive Events in detail?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other event absolutely cannot occur. They have no common outcomes.
Simple Example
Quick Example
Imagine you are flipping a coin. The event 'getting a head' and the event 'getting a tail' are mutually exclusive. You cannot get both a head and a tail on a single flip.
Worked Example
Step-by-Step
Let's say we have a bag of marbles: 5 red, 3 blue, and 2 green. We pick one marble at random.
---Step 1: Define Event A as 'picking a red marble'.
---Step 2: Define Event B as 'picking a blue marble'.
---Step 3: Can you pick a marble that is both red AND blue at the same time from this bag?
---Step 4: No, a single marble cannot be both red and blue.
---Step 5: Since Event A (picking red) and Event B (picking blue) cannot happen simultaneously, they are mutually exclusive events.
---Answer: Picking a red marble and picking a blue marble are mutually exclusive events.
Why It Matters
Understanding mutually exclusive events is crucial in AI/ML for building reliable decision models, in FinTech for risk assessment, and in Medicine for understanding disease outcomes. Engineers use this concept to design systems where certain failures cannot occur together, ensuring safety and efficiency.
Common Mistakes
MISTAKE: Thinking 'not mutually exclusive' means they always happen together. | CORRECTION: If events are not mutually exclusive, it simply means they CAN happen together, not that they always will. They might overlap sometimes.
MISTAKE: Confusing mutually exclusive with independent events. | CORRECTION: Mutually exclusive means they cannot happen together. Independent means one event happening does not affect the probability of the other. These are different concepts.
MISTAKE: Assuming all events are either mutually exclusive or not. | CORRECTION: Always check if there's any overlap in their outcomes. If the intersection of their outcomes is empty, they are mutually exclusive.
Practice Questions
Try It Yourself
QUESTION: Are the events 'rolling an even number' and 'rolling an odd number' on a standard six-sided dice mutually exclusive? | ANSWER: Yes
QUESTION: In a class of 40 students, 25 play cricket and 20 play football. Are the events 'a student plays cricket' and 'a student plays football' mutually exclusive? Explain. | ANSWER: No. Since 25 + 20 = 45, which is more than 40 students, it means some students must be playing both games. So, there is an overlap, and they are not mutually exclusive.
QUESTION: You are drawing a single card from a standard deck of 52 cards. Are the events 'drawing a King' and 'drawing a red card' mutually exclusive? | ANSWER: No. You can draw a King of Hearts or a King of Diamonds, which are both Kings and red cards. So, they can happen together.
MCQ
Quick Quiz
Which of the following pairs of events is mutually exclusive?
Eating dosa for breakfast and drinking coffee for breakfast.
A student passing an exam and a student failing the same exam.
It raining today and it being cloudy today.
Picking a red ball and picking a blue ball from a bag containing only red and blue balls, if you pick two balls.
The Correct Answer Is:
B
A student cannot both pass and fail the same exam at the same time, making these events mutually exclusive. Options A and C can happen together. Option D involves picking two balls, so you can pick one red and one blue.
Real World Connection
In the Real World
When using a payment app like UPI, you either successfully complete a transaction or the transaction fails. These two outcomes are mutually exclusive – a payment cannot be both successful and failed at the same time. This ensures clear financial records and reliable transactions.
Key Vocabulary
Key Terms
EVENT: A specific outcome or a set of outcomes in an experiment. | OUTCOME: A single possible result of an experiment. | SAMPLE SPACE: The set of all possible outcomes of an experiment. | INTERSECTION: The outcomes that are common to two or more events.
What's Next
What to Learn Next
Now that you understand mutually exclusive events, you're ready to explore 'Independent Events'. Learning about independent events will help you understand how events can happen without influencing each other, which is another crucial concept in probability.
