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 is 1/66 or approximately 1.52%.

Step-by-step solution

Determine the total number of combinations for selecting three houses from the row of 12 houses.

Using the combination formula, we can determine the total number of ways to choose three houses from the 12 houses: C(n, k) = C(12, 3) = \frac{12!}{(12-3)!3!} = \frac{12!}{9!3!} Here, n = 12 (the total number of houses) and k = 3 (the number of houses we want to choose).

Compute the value of C(12,3).

Now, we can calculate C(12,3): C(12,3) = \frac{12!}{9!3!} = \frac{12×11×10}{3×2×1} = 12×11×5 = 660 So, 660 is the total number of combinations for selecting any three houses from the row of 12 houses.

Calculate the number of ways to select three adjacent houses from the row of 12 houses.

There are 12 houses in a row, and we need to find three adjacent houses. Let's denote the first house of the three adjacent houses as H_n. When H_n is the first house, H_{n+1} and H_{n+2} will be the other two adjacent houses. To count the number of ways for selecting three adjacent houses, we can consider the position of H_n: If H_1 is the first house, then H_2 and H_3 will be adjacent If H_2 is the first house, then H_3 and H_4 will be adjacent ... If H_10 is the first house, then H_11 and H_12 will be adjacent So, there are a total of 10 ways to select three adjacent houses.

Identify the total probability of selecting three adjacent houses from the row of 12 houses.

Now, we can calculate the probability of choosing three adjacent houses. It is the ratio of the number of favorable outcomes (choosing three adjacent houses) to the total number of possible outcomes (all combinations): Probability = \frac{Number\ of\ Favorable\ Outcomes}{Total\ Number\ of\ Possible\ Outcomes} = \frac{10}{660}

Simplify the probability to find the final answer.

Lastly, we simplify the probability. We can divide both the numerator and denominator by 10 to find the simplest form: Probability = \frac{10}{660} = \frac{1}{66} So, there is a 1/66 (∼0.0152 or 1.52%) probability of selecting three adjacent houses from the row of 12 houses. To determine if there is reason to believe the virus is contagious, it would be important to consider other factors such as how the virus is transmitted, any patterns of symptoms, and any documented cases of this virus spreading within close proximity. The 1.52% probability that three adjacent houses would be selected can be seen as a low probability; however, it is not enough evidence by itself to determine the contagiousness of the virus.

Key concepts

Understanding Combinations

In probability and statistics, combinations are used to determine how many ways you can choose a certain number of items from a larger set without considering the order. This is done using the combination formula:

• The formula is expressed as \( C(n, k) = \frac{n!}{(n-k)!k!} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
• The \(!\) symbol refers to a factorial, which is the product of all positive integers up to that number. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In our scenario involving 12 houses, we need to determine in how many ways we can select 3 houses out of 12. By applying the combination formula, we calculate \( C(12, 3) = \frac{12!}{9!3!} = 660 \). This means there are 660 possible combinations to choose any 3 houses from a row of 12 houses.

Determining Adjacent Houses

Adjacent houses refer to houses that are next to one another in a sequence. When selecting these adjacent houses from a lineup, their arrangement or order significantly matters because they must be in a row without gaps. To find the number of ways to pick 3 adjacent houses in a sequence of 12 houses, consider that the first house in every group can start from house \( H_1 \) to house \( H_{10} \) since three houses are chosen at a time. Think of it like this:

• If starting at \( H_1 \), the adjacent houses are \( H_1, H_2, \text{and} H_3 \).
• If starting at \( H_2 \), the adjacent houses are \( H_2, H_3, \text{and} H_4 \).
• The pattern continues until \( H_{10}, H_{11}, \text{and} H_{12} \), making it evident that there are 10 possibilities.

Thus, there are 10 ways to select 3 adjacent houses from a row of 12 houses.

Calculating Probability

Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain). In this exercise, we calculate the probability of selecting 3 adjacent houses randomly from a row of 12. Here's the formula to use:

• Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \).

From the previous calculations, we know there are 10 favorable outcomes (selecting 3 adjacent houses) and 660 total outcomes (all combinations of 3 houses from 12).Substituting these values in the formula:

• Probability = \( \frac{10}{660} = \frac{1}{66} \), which indicates there is a 1 in 66 chance, or approximately 1.52% probability, of randomly selecting 3 adjacent houses.

This small probability shows choosing adjacent houses by chance is relatively unlikely.

Exploring the Contagion Hypothesis

The contagion hypothesis relates to understanding how a virus might spread between houses in close proximity. In the context of adjacent houses, it suggests that the virus may more likely spread from one house to the neighboring ones, assuming contagion is the transmission mechanism. Evaluating whether a virus is contagious may require more than just analyzing the pattern of infection across adjacent houses. Important considerations include:

• The way the virus is transmitted, such as through air, direct contact, or surfaces.
• Observed symptoms that might indicate transmission patterns and the virus's behavior.
• Data or evidence from other documented cases where the virus spread.

While a 1.52% probability of selecting 3 adjacent houses at random from 12 is quite low, it isn't conclusive on its own about the virus's contagiousness. Further scientific investigation would be necessary to substantiate or refute the hypothesis.