What are Exhaustive Events? | Simple Explanation for Class 12

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What are Exhaustive 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?

Exhaustive events are a set of events in a random experiment such that at least one of them is guaranteed to occur when the experiment is performed. In simpler terms, if you list all possible outcomes, exhaustive events cover every single one, leaving no outcome out.

Simple Example

Quick Example

Imagine you are playing a game of Ludo and roll a single dice. The events 'getting an even number' (2, 4, 6) and 'getting an odd number' (1, 3, 5) are exhaustive. Why? Because when you roll the dice, you are guaranteed to get either an even or an odd number; there's no other possibility.

Worked Example

Step-by-Step

Let's say we have a spinner with three colours: Red (R), Green (G), and Blue (B). We spin it once.
---Step 1: Identify the sample space (all possible outcomes). Here, S = {R, G, B}.
---Step 2: Define a set of events. Let Event A be 'getting Red or Green' (A = {R, G}). Let Event B be 'getting Blue' (B = {B}).
---Step 3: Check if the union of these events covers the entire sample space. Union of A and B (A ∪ B) = {R, G} ∪ {B} = {R, G, B}.
---Step 4: Compare A ∪ B with the sample space S. Since A ∪ B = S, events A and B are exhaustive.
Answer: Yes, the events 'getting Red or Green' and 'getting Blue' are exhaustive.

Why It Matters

Understanding exhaustive events helps in predicting outcomes and calculating probabilities, which is crucial in fields like AI/ML for making smart decisions or in FinTech for risk analysis. Engineers use this to design reliable systems, and doctors use it to understand all possible patient responses to treatments.

Common Mistakes

MISTAKE: Confusing exhaustive events with mutually exclusive events. | CORRECTION: Exhaustive events cover all possibilities (their union is the sample space). Mutually exclusive events cannot happen at the same time (their intersection is empty). An event can be both, one, or neither.

MISTAKE: Forgetting to include all possible outcomes when defining events. | CORRECTION: Always start by clearly listing the entire sample space (all possible results) of the experiment before defining your events. Then ensure the union of your events matches this sample space.

MISTAKE: Thinking only two events can be exhaustive. | CORRECTION: You can have two, three, or many events that are exhaustive, as long as their combined outcomes cover the entire sample space.

Practice Questions

Try It Yourself

QUESTION: You flip a coin. Let Event A be 'getting Heads'. Let Event B be 'getting Tails'. Are these events exhaustive? | ANSWER: Yes

QUESTION: A bag contains 5 red balls and 3 blue balls. You pick one ball. Let Event E1 be 'picking a red ball'. Let Event E2 be 'picking a blue ball'. Let Event E3 be 'picking a green ball'. Are E1, E2, and E3 exhaustive? | ANSWER: No (because there are no green balls, but E3 is defined as picking one. The union of E1 and E2 covers all actual possibilities, but E3 introduces an impossible outcome, making the set {E1, E2, E3} not strictly exhaustive in the sense that E3 itself cannot occur and doesn't add to covering the sample space.)

QUESTION: In a cricket match, what events would be exhaustive for the outcome of a single ball bowled by a bowler? | ANSWER: The events 'runs scored (0, 1, 2, 3, 4, 5, 6)', 'wicket taken', and 'extra (wide, no-ball)' are exhaustive. (You could also simplify to 'wicket', 'no wicket but runs', 'no wicket and no runs', 'extras' if you want broader categories).

MCQ

Quick Quiz

Which of the following sets of events is exhaustive when rolling a standard six-sided dice?

Getting a 1 or getting a 6

Getting an even number or getting an odd number

Getting a number less than 3 or getting a number greater than 4

Getting a prime number or getting an even number

The Correct Answer Is:

Option B covers all possible outcomes (1, 2, 3, 4, 5, 6) because every number is either even or odd. The other options leave out some possible outcomes.

Real World Connection

In the Real World

Think about online food delivery apps like Zomato or Swiggy. When you place an order, the events 'order delivered successfully', 'order cancelled by restaurant', and 'order cancelled by customer' are exhaustive for the status of your order. Data scientists use this concept to ensure all possible outcomes are considered when designing algorithms for logistics or customer service.

Key Vocabulary

Key Terms

SAMPLE SPACE: All possible outcomes of an experiment. | EVENT: A specific outcome or a set of outcomes from an experiment. | UNION OF EVENTS: The collection of all outcomes that are in at least one of the events. | MUTUALLY EXCLUSIVE EVENTS: Events that cannot happen at the same time.

What's Next

What to Learn Next

Now that you understand exhaustive events, you should explore 'Mutually Exclusive Events' and 'Independent Events'. These concepts often go hand-in-hand and are essential for mastering probability and making accurate predictions in various real-world scenarios.