Calculation of Probability for Exhaustive Events
Calculating the probability for exhaustive events involves determining the likelihood of each individual event occurring and then using the concept of exhaustiveness to find the probability of at least one of the events happening. The steps to calculate probability for exhaustive events are given below:
Step 1: Determine Individual Probabilities
Calculate the probability of each individual event occurring. This can be done by dividing the number of favorable outcomes (for each event) by the total number of possible outcomes in the sample space.
Step 2: Verify Exhaustiveness
Ensure that the events are collectively exhaustive, meaning that together they cover all possible outcomes.
Step 3: Use the Principle of Exhaustiveness
Since the events are collectively exhaustive, the probability of at least one of them occurring is equal to 1 (or 100%). This is because if one event doesn’t happen, then another event in the set must happen. Therefore, the sum of the probabilities of all exhaustive events equals 1.
Example: Consider rolling a fair six-sided die. The possible outcomes are numbers 1 through 6.
- Events A: Rolling an odd number {1, 3, 5}
- Events B: Rolling an even number {2, 4, 6}
Solution:
Step 1: Calculate Individual Probabilities
- Probability of Event A:
P(A) = 3/6 = 1/2
- Probability of Event B:
P(B) = 3/6 =1/2
Step 2: Verify Exhaustiveness
Together, Events A and B cover all possible outcomes (1 through 6), making them collectively exhaustive.
Step 3: Use the Principle of Exhaustiveness
Since the events are collectively exhaustive, the probability of at least one of them occurring is 1 (or 100%).
So, in this example, the probability of rolling either an odd number or an even number (or both) on a fair six-sided die is 1. This means that you are guaranteed to roll an odd or an even number when you roll the die.
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Exhaustive Events
Exhaustive Events are a set of events where at least one of the events must occur while performing an experiment. Exhaustive events are a set of events whose union makes up the complete sample space of the experiment.
In this article, we will understand the meaning of exhaustive events, its definition, Venn diagram of exhaustive events, collective exhaustive events, and examples of exhaustive events.
Table of Content
- What are Exhaustive Events?
- Exhaustive Event Venn Diagram
- Collectively Exhaustive Events
- Examples of Exhaustive Events
- Calculation of Probability for Exhaustive Events