Introduction to the Problem: This article delves into the probability calculation of selecting two consecutive numbers from a set of numbers ranging from 1 to 20, covering different scenarios such as with or without replacement. We will explore the mathematical steps involved and discuss the nuances of probability theory.

Setting the Stage

We are interested in finding the probability that two randomly chosen numbers from the set {1, 2, ..., 20} are consecutive. Let's analyze this problem from first principles.

Determining the Total Number of Outcomes

Without Replacement:

When selecting two numbers without replacement, the total number of possible outcomes is calculated using combinations. The binomial coefficient 20C(2) represents the number of ways to choose 2 numbers out of 20 without regard to order.

[20 , C , 2 frac{20 times 19}{2} 190]

With Replacement:

When selection is with replacement, each number can be chosen twice, leading to a different total number of outcomes. The total number of pairs is the product of the number of choices for each selection.

[20 times 20 400]

Calculating the Number of Favorable Outcomes

Without Replacement:

Two consecutive numbers from the set can be paired as follows: 1-2, 2-3, ..., 19-20. There are 19 such pairs.

With Replacement:

The pairs are still the same 19 but we must now consider both the pair (1,2) and (2,1) as distinct. Thus, there are 2 * 19 38 such pairs.

Probability Calculation

Without Replacement

The probability of selecting two consecutive numbers without replacement is:

[frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{19}{190} frac{1}{10}]

This is the correct probability for the without replacement scenario.

With Replacement

For the with replacement scenario, the total number of pairs is 400, and the number of pairs of consecutive numbers is 38 (consider both (1,2) and (2,1)). Thus, the probability is:

[frac{38}{400} frac{19}{200} 0.095 9.5%]

Further Analysis

It's interesting to note that even though the number of pairs of consecutive numbers remains the same (19) in both scenarios, the probability changes due to the nature of the selection (with or without replacement).

Conclusion: The probability that two chosen numbers are consecutive numbers in the set {1, 2, ..., 20} is 1/10 when selections are made without replacement. However, the probability decreases to 9.5% when selections are made with replacement.

Keywords: consecutive numbers, probability calculation, combinatorics