Question

A certain virus afflicted the families in three adjacent houses in a row of 12 houses. If houses were randomly chosen from a row of 12 houses, what is the probability that the three houses would be adjacent? Is there reason to believe that this virus is contagious?

Short answer

Answer: The probability that three randomly chosen houses are adjacent in a row of 12 houses is 1/22. However, this information alone is not sufficient to determine if the virus is contagious. Further analysis and investigation are needed, such as contact tracing, monitoring the spread of the virus in other areas, and studying the characteristics of the virus itself.

Step-by-step solution

Determine the total number of ways to choose three houses

There are a total of 12 houses and we need to choose 3 of them. The number of ways to choose 3 houses from 12 is given by the combination, which can be calculated using the formula: C(n, k) = n! / (k!(n-k)!) Here, n = 12 (total number of houses), and k = 3 (number of houses to choose). C(12, 3) = 12! / (3! * (12-3)!) = 220

Determine the number of ways to choose three adjacent houses

To find the ways to choose three adjacent houses, we can imagine that there is a "moving window" of size 3 that we slide along the row of 12 houses. In each position, the three houses covered by this window are adjacent. Since there are 12 houses in the row and the window has a size of 3, there are: (12 - 3) + 1 = 10 possible positions of this window.

Calculate the probability

Now that we have the total number of ways to choose three houses and the number of ways to choose three adjacent houses, we can calculate the probability by dividing the number of ways to choose three adjacent houses by the total number of ways to choose three houses. So, the probability is: Probability = (Number of ways to choose three adjacent houses) / (Total number of ways to choose three houses) Probability = 10 / 220 = 1 / 22

Determine if there is enough evidence to believe the virus is contagious

The probability that three randomly chosen houses are adjacent is 1/22, which is quite low. However, we cannot conclude that the virus is contagious or not based on this probability alone. The spread of the virus could be affected by several other factors like the presence of a common source (like contaminated water supply or food), environmental factors, or the behavior of the inhabitants of those houses. To determine if the virus is contagious, more information and analysis are needed, such as contact tracing, monitoring the spread of the virus in other areas, and studying the characteristics of the virus itself.

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and is widely utilized in computer science.

When solving the probability adjacent houses problem, combinatorics helps us answer the question: In how many ways can we choose three houses out of twelve? The formula used in this context is for combinations, denoted as C(n, k), which represents the number of ways to choose k elements from a set of n distinct elements without considering the order of selection. The formula for calculating a combination is: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]
In our example, we had to calculate C(12, 3), representing the number of ways to select 3 houses from 12 distinct houses, which equaled 220. Understanding this concept is fundamental as it serves as a basis for computing the overall probability of the event.

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random phenomena and dealing with the likelihood that a particular event will occur. It provides a quantitative description of the chance or probability associated with various outcomes. The probability of an event is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

To apply probability theory to our problem, we consider all possible outcomes of choosing three houses (the 'sample space') and the event of interest (choosing three adjacent houses). By dividing the number of favorable outcomes by the number of possible outcomes, we get the probability:\[ \text{Probability} = \frac{\text{Number of ways to choose three adjacent houses}}{\text{Total number of ways to choose three houses}} \]For the adjacent houses problem, we found there were 10 ways (favorable outcomes) to choose three adjacent houses out of the 220 ways (total possible outcomes) to choose any three houses, yielding a probability of 1/22. Probability theory guides us to quantify the chance of the virus spreading by natural random selection of houses, which aids in the evaluation of contagion risks.

Statistics

Statistics is the discipline that concerns the collection, analysis, interpretation, presentation, and organization of data. In dealing with the probability adjacent houses problem, we use statistics to interpret the results we obtained through combinatorics and probability theory, and to understand what these results tell us about the real-world situation at hand.

In the described problem, we obtained a probability of 1/22 for three adjacent houses to be chosen at random. However, statistical reasoning requires us to consider this result in the context of additional data and information. For instance, examining the rate of virus spread in comparison to what would be expected under random circumstances is a statistical approach that might provide insight into whether the virus is contagious. To properly assess this, one might conduct hypothesis tests or analyze further data from contact tracing or environmental studies. Therefore, beyond just calculating probability, statistics is crucial in forming reliable conclusions based on numerical data.