We appreciate your visit to The director of health services is concerned about a possible flu outbreak at her college She surveyed 100 randomly selected residents from the college s. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
- Define events: $A$ is having a flu shot, $B$ is being male.
- Use conditional probability formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
- Calculate $P(A \cap B) = \frac{39}{100}$ and $P(B) = \frac{51}{100}$.
- Substitute and simplify: $P(A|B) = \frac{39}{51} = \boxed{\frac{13}{17}}$.
### Explanation
1. Understand the problem and provided data
We are given a table that summarizes the results of a survey about flu shots among college dormitory residents. The goal is to find the probability that a randomly chosen dormitory resident has had a flu shot, given that the resident is male.
2. Define events and conditional probability formula
Let $A$ be the event that a resident has had a flu shot, and let $B$ be the event that a resident is male. We want to find the conditional probability $P(A|B)$, which is the probability that a resident has had a flu shot given that the resident is male. The formula for conditional probability is: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
3. Calculate $P(A
cap B)$
$P(A \cap B)$ is the probability that a resident is male and has had a flu shot. From the table, there are 39 male residents who had a flu shot out of the 100 total residents. So, $$P(A \cap B) = \frac{39}{100}$$
4. Calculate $P(B)$
$P(B)$ is the probability that a resident is male. From the table, there are 51 male residents out of the 100 total residents. So, $$P(B) = \frac{51}{100}$$
5. Apply the conditional probability formula
Now, we substitute these values into the conditional probability formula:$$P(A|B) = \frac{\frac{39}{100}}{\frac{51}{100}} = \frac{39}{51}$$
6. Simplify and conclude
We simplify the fraction $\frac{39}{51}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:$$\frac{39}{51} = \frac{39 \div 3}{51 \div 3} = \frac{13}{17}$$Therefore, the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, is $\frac{13}{17}$.
### Examples
Understanding conditional probabilities is very useful in real life. For example, in medical testing, it helps doctors determine the probability of a patient having a disease given a positive test result, considering the test's accuracy and the prevalence of the disease in the population.
- Use conditional probability formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
- Calculate $P(A \cap B) = \frac{39}{100}$ and $P(B) = \frac{51}{100}$.
- Substitute and simplify: $P(A|B) = \frac{39}{51} = \boxed{\frac{13}{17}}$.
### Explanation
1. Understand the problem and provided data
We are given a table that summarizes the results of a survey about flu shots among college dormitory residents. The goal is to find the probability that a randomly chosen dormitory resident has had a flu shot, given that the resident is male.
2. Define events and conditional probability formula
Let $A$ be the event that a resident has had a flu shot, and let $B$ be the event that a resident is male. We want to find the conditional probability $P(A|B)$, which is the probability that a resident has had a flu shot given that the resident is male. The formula for conditional probability is: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
3. Calculate $P(A
cap B)$
$P(A \cap B)$ is the probability that a resident is male and has had a flu shot. From the table, there are 39 male residents who had a flu shot out of the 100 total residents. So, $$P(A \cap B) = \frac{39}{100}$$
4. Calculate $P(B)$
$P(B)$ is the probability that a resident is male. From the table, there are 51 male residents out of the 100 total residents. So, $$P(B) = \frac{51}{100}$$
5. Apply the conditional probability formula
Now, we substitute these values into the conditional probability formula:$$P(A|B) = \frac{\frac{39}{100}}{\frac{51}{100}} = \frac{39}{51}$$
6. Simplify and conclude
We simplify the fraction $\frac{39}{51}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:$$\frac{39}{51} = \frac{39 \div 3}{51 \div 3} = \frac{13}{17}$$Therefore, the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, is $\frac{13}{17}$.
### Examples
Understanding conditional probabilities is very useful in real life. For example, in medical testing, it helps doctors determine the probability of a patient having a disease given a positive test result, considering the test's accuracy and the prevalence of the disease in the population.
Thanks for taking the time to read The director of health services is concerned about a possible flu outbreak at her college She surveyed 100 randomly selected residents from the college s. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
