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Answer :
Let's solve the problem step-by-step to determine the required probability.
### Step-by-Step Solution:
1. Identify the Given Information:
- We have a table that provides data based on whether dormitory residents at college have had a flu shot or not, segmented by gender.
- The relevant figures from the table are:
- Number of males who have had a flu shot: [tex]\(39\)[/tex]
- Total number of male residents: [tex]\(51\)[/tex]
2. Understand the Question:
- We are asked to find the probability that a randomly chosen dormitory resident has had a flu shot, given that the resident is male.
3. Concept of Conditional Probability:
- The probability of an event [tex]\(A\)[/tex] happening, given that [tex]\(B\)[/tex] has already occurred, is represented as [tex]\(P(A|B)\)[/tex].
- In this case, we want [tex]\(P(\text{Had Flu Shot}|\text{Male})\)[/tex].
4. Apply the Formula for Conditional Probability:
- The formula for conditional probability is:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{\text{Number of males who had a flu shot}}{\text{Total number of males}}
\][/tex]
- Substituting the numbers from the given data into the formula:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{39}{51}
\][/tex]
5. Calculate the Result:
- Dividing the numerator by the denominator yields the probability:
[tex]\[
\frac{39}{51} \approx 0.7647
\][/tex]
6. Conclusion:
- Therefore, the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, is approximately [tex]\(0.7647\)[/tex].
The answer is:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) \approx 0.7647
\][/tex]
Hence, the required probability is about [tex]\(0.7647\)[/tex] or [tex]\(76.47\%\)[/tex].
### Step-by-Step Solution:
1. Identify the Given Information:
- We have a table that provides data based on whether dormitory residents at college have had a flu shot or not, segmented by gender.
- The relevant figures from the table are:
- Number of males who have had a flu shot: [tex]\(39\)[/tex]
- Total number of male residents: [tex]\(51\)[/tex]
2. Understand the Question:
- We are asked to find the probability that a randomly chosen dormitory resident has had a flu shot, given that the resident is male.
3. Concept of Conditional Probability:
- The probability of an event [tex]\(A\)[/tex] happening, given that [tex]\(B\)[/tex] has already occurred, is represented as [tex]\(P(A|B)\)[/tex].
- In this case, we want [tex]\(P(\text{Had Flu Shot}|\text{Male})\)[/tex].
4. Apply the Formula for Conditional Probability:
- The formula for conditional probability is:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{\text{Number of males who had a flu shot}}{\text{Total number of males}}
\][/tex]
- Substituting the numbers from the given data into the formula:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{39}{51}
\][/tex]
5. Calculate the Result:
- Dividing the numerator by the denominator yields the probability:
[tex]\[
\frac{39}{51} \approx 0.7647
\][/tex]
6. Conclusion:
- Therefore, the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, is approximately [tex]\(0.7647\)[/tex].
The answer is:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) \approx 0.7647
\][/tex]
Hence, the required probability is about [tex]\(0.7647\)[/tex] or [tex]\(76.47\%\)[/tex].
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