# COVID-19 in a Small Town: Probability of Infection

• MHB
• laprec
In summary, the data showed that 22% of people in a small town were infected with COVID-19. A random sample of six residents from this town was selected. The probability that exactly two of these residents were infected is 15*(0.22)^2*(0.78)^4. The probability that at most one of these residents was infected is (0.78)^6 + 6*0.22*(0.78)^5.
laprec
Kindly assist with these questions:
Data showed that 22% of people in a small town was infected with the COVID-19 virus. A random sample of six residents from this town was selected.
1) What is the probability that exactly two of these residents was infected?
2) What is the probability that at most 1 of these residents was infected?
Thank you

If the probability a given person has the virus is p then the probability that person does not have it is 1- p. First imagine putting the six people in a given order. The probability the first person has the virus is p, the probability the next person has the virus is p, the probability the third person does NOT have the virus is 1- p, the probability the fourth person does not is1-p, and the probability the fifth and sixth persons do not is 1-p.

The probability the first two people have the virus and the other four do not, in that order, is $p^2(1-p)^4$.

There are $\frac{6!}{2!4!}= \frac{6(5)}{2}= 15$ different orders of those two people who have the virus and four who do not so the probability two of a random six people have the diease and four do not is $15p^2(1- p)^4$.

In your problem, of course, p= 0.22.

"At most one" means "either one or none". By the same argument as above, the probability none of the people has the virus is $(1- p)^6$ and the probability exactly one has it is $6p(1-p)^5$. Since "none" and "exactly one" are "mutually exclusive", the probability of "at most one" is the sum of those.

## 1. What is the probability of getting infected with COVID-19 in a small town?

The probability of getting infected with COVID-19 in a small town depends on various factors such as population density, adherence to safety measures, and the number of active cases in the town. It is difficult to determine an exact probability, but following safety guidelines and avoiding high-risk activities can decrease the chances of infection.

## 2. How does the probability of infection in a small town compare to a larger city?

The probability of infection in a small town may be lower compared to a larger city due to a smaller population and less crowded public spaces. However, it is important to note that the risk of infection can still be high in a small town if safety measures are not followed.

## 3. Is it safe to visit a small town during the COVID-19 pandemic?

The safety of visiting a small town during the COVID-19 pandemic depends on the current situation of the town and the precautions taken by both visitors and residents. It is important to research the number of active cases and safety guidelines in place before visiting a small town.

## 4. Can COVID-19 spread quickly in a small town?

COVID-19 can spread quickly in a small town if safety measures are not followed. The close-knit community and limited healthcare resources in small towns can make it challenging to control the spread of the virus. It is important for everyone to practice social distancing, wear masks, and follow other safety guidelines to prevent the spread of the virus.

## 5. How can a small town prevent a COVID-19 outbreak?

A small town can prevent a COVID-19 outbreak by implementing safety measures such as wearing masks, practicing social distancing, and regularly sanitizing public spaces. It is also important for residents to stay informed about the current situation and follow guidelines from health authorities. Testing and contact tracing can also help identify and isolate potential cases to prevent an outbreak.

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