Probability of Selecting Four Odd Numbers from 20 Consecutive Cards with Replacement
In this article, we will explore a problem in probability theory that involves selecting four odd numbers from twenty consecutive cards with replacement. Understanding such problems is crucial for individuals interested in data analysis, statistics, and those who wish to improve their problem-solving skills. We'll break down the problem step-by-step, ensuring clarity and providing a clear solution.
The Problem Statement
The problem states that we have twenty index cards numbered consecutively from 1 through 20. We are to determine the probability of selecting four odd numbers if replacement is allowed in the selection process. The selection is made in four steps, and with replacement, the same card can be drawn more than once.
The Solution Approach
To solve this problem, we need to understand the fundamental principles of probability and combinations. First, we must identify the total number of odd numbers among the 20 cards and then calculate the probability of drawing an odd number in each step. Finally, we'll combine these probabilities to find the overall probability of drawing four odd numbers.
Identifying Odd Numbers
The odd numbers between 1 and 20 are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. Hence, there are 10 odd numbers among the 20 cards. This gives us a favorable outcome of 10 out of 20 cards being odd.
Calculating the Probability for One Draw
The probability of drawing an odd number in one draw is the number of odd numbers divided by the total number of cards:
Probability of one odd number 10/20 1/2
Calculating the Probability for Four Draws with Replacement
Since the drawing is done with replacement, the probability of drawing an odd number in each draw remains the same, which is 1/2. To find the probability of drawing four odd numbers in four draws, we need to multiply the probabilities of each individual draw:
Probability of four odd numbers (1/2) x (1/2) x (1/2) x (1/2) (1/2)^4 1/16
Verification of the Solution
In the provided solution, the probability is calculated as 10^4/20^4, which simplifies to 1/16. This aligns with our step-by-step reasoning and confirms the accuracy of the solution.
Conclusion
Understanding and solving problems like the one presented here not only enhances your probability skills but also helps in various real-world applications, such as statistical analysis and data-driven decision-making. Whether you're a student, an academic, or a professional in fields that rely on data, mastering these concepts is invaluable.
Further Exploration
For those interested in delving deeper into probability and its applications, we recommend exploring the following topics:
Combinatorics and permutations Conditional probability Bayesian inference and its applications Probability distributions and their propertiesBy understanding these concepts, you can approach similar problems and extend your knowledge to more complex scenarios.