What are Mutually Inclusive Events?

S7-SA3-0384

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Mutually inclusive events are events that can happen at the same time. This means they share at least one common outcome or possibility. They are not exclusive of each other.

Simple Example

Quick Example

Imagine you are picking a number from 1 to 10. Event A is picking an even number (2, 4, 6, 8, 10). Event B is picking a number greater than 5 (6, 7, 8, 9, 10). These events are mutually inclusive because numbers like 6, 8, and 10 are common to both events.

Worked Example

Step-by-Step

PROBLEM: In a class of 30 students, 15 students play Cricket (C) and 10 students play Football (F). 5 students play both Cricket and Football. What is the probability that a randomly chosen student plays Cricket OR Football?

Step 1: Identify the events and given probabilities.
P(C) = Probability of playing Cricket = Number of students playing Cricket / Total students = 15/30 = 0.5
P(F) = Probability of playing Football = Number of students playing Football / Total students = 10/30 = 0.33 (approx)
P(C and F) = Probability of playing both Cricket and Football = Number of students playing both / Total students = 5/30 = 0.167 (approx)
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Step 2: Recognize that these are mutually inclusive events because students can play both sports.
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Step 3: Use the formula for mutually inclusive events: P(C or F) = P(C) + P(F) - P(C and F).
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Step 4: Substitute the values into the formula.
P(C or F) = (15/30) + (10/30) - (5/30)
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Step 5: Calculate the sum.
P(C or F) = (15 + 10 - 5) / 30 = 20 / 30
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Step 6: Simplify the fraction.
P(C or F) = 2/3

Answer: The probability that a randomly chosen student plays Cricket OR Football is 2/3 (or approximately 0.67).

Why It Matters

Understanding mutually inclusive events helps in predicting outcomes in many fields. For example, in AI/ML, it helps build models that understand overlapping data. In FinTech, it's used to assess risks where multiple factors can occur together, helping banks make smarter decisions about loans and investments.

Common Mistakes

MISTAKE: Assuming all events are mutually exclusive and simply adding their probabilities. | CORRECTION: Always check if there's an overlap (common outcomes). If there is, subtract the probability of the overlap to avoid double-counting.

MISTAKE: Forgetting to subtract the probability of the intersection (P(A and B)) when using the formula P(A or B) = P(A) + P(B). | CORRECTION: Remember the formula for mutually inclusive events is P(A or B) = P(A) + P(B) - P(A and B). The '- P(A and B)' part is crucial.

MISTAKE: Confusing 'mutually inclusive' with 'independent events'. | CORRECTION: Mutually inclusive means they can happen together. Independent means one event happening doesn't affect the probability of the other happening. They are different concepts.

Practice Questions

Try It Yourself

QUESTION: A card is drawn from a standard deck of 52 cards. What is the probability of drawing a King OR a Heart? | ANSWER: P(King or Heart) = P(King) + P(Heart) - P(King of Hearts) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13

QUESTION: In a group of 40 tourists, 25 speak Hindi, 18 speak English, and 10 speak both. What is the probability that a randomly selected tourist speaks Hindi OR English? | ANSWER: P(Hindi or English) = P(Hindi) + P(English) - P(Both) = 25/40 + 18/40 - 10/40 = (25+18-10)/40 = 33/40

QUESTION: A survey found that 60% of people like 'chai', 45% like 'coffee', and 20% like both. If a person is chosen randomly, what is the probability that they like chai OR coffee? | ANSWER: P(Chai or Coffee) = P(Chai) + P(Coffee) - P(Both) = 0.60 + 0.45 - 0.20 = 1.05 - 0.20 = 0.85 (or 85%)

MCQ

Quick Quiz

Which of the following describes mutually inclusive events?

Events that cannot happen at the same time.

Events that share one or more common outcomes.

Events where the outcome of one affects the other.

Events that are always equally likely to occur.

The Correct Answer Is:

Mutually inclusive events are defined as events that can happen together, meaning they have at least one common outcome. Option A describes mutually exclusive events. Options C and D are incorrect definitions.

Real World Connection

In the Real World

Imagine an app like Zomato or Swiggy recommending restaurants. They might consider restaurants that are 'within 5 km' (Event A) AND 'offer a discount' (Event B). Since a restaurant can be both within 5 km and offer a discount, these are mutually inclusive events. The app uses this understanding to show you the best options, avoiding double-counting places that fit both criteria.

Key Vocabulary

Key Terms

PROBABILITY: The chance of an event happening. | EVENT: A specific outcome or set of outcomes. | INTERSECTION: The set of outcomes common to two or more events. | UNION: The set of all outcomes in two or more events combined. | OUTCOME: A possible result of an experiment or situation.

What's Next

What to Learn Next

Now that you understand mutually inclusive events, you should explore 'Mutually Exclusive Events'. This will help you clearly differentiate between events that can happen together and those that cannot, which is key for solving more complex probability problems.