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Answer :
Sure, let's solve this question step-by-step:
1. Understand the Given Data:
- A total of 100 residents were surveyed.
- Out of these,
- 39 males had a flu shot.
- 12 males did not have a flu shot.
- So, there are 51 males in total (39 + 12).
2. Identify the Question:
- We are asked to find the probability that a dormitory resident chosen at random has had a flu shot, given that he is male.
- This is conditional probability.
3. Define Conditional Probability:
- Conditional probability [tex]\( P(A|B) \)[/tex] is the probability of event [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has occurred.
- Here, [tex]\( A \)[/tex] is the event "having had a flu shot" and [tex]\( B \)[/tex] is the event "being male".
4. Calculate the Required Probability:
- We need to find [tex]\( P(\text{Had Flu Shot}|\text{Male}) \)[/tex].
- From the data, we know:
- The total number of males is 51.
- The number of males who had a flu shot is 39.
- The probability that a randomly chosen male has had a flu shot can be calculated as:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{\text{Number of males who had a flu shot}}{\text{Total number of males}} = \frac{39}{51}
\][/tex]
5. Simplify the Fraction:
- Simplifying [tex]\(\frac{39}{51}\)[/tex] might give us a clearer fraction, but here we use the result known:
[tex]\[
\frac{39}{51} \approx 0.7647
\][/tex]
6. State the Final Answer:
- Therefore, the probability that a dormitory resident chosen at random has had a flu shot, given that he is male, is approximately 0.765 or 76.47%.
So, the probability is approximately [tex]\( 0.7647058823529411 \)[/tex].
1. Understand the Given Data:
- A total of 100 residents were surveyed.
- Out of these,
- 39 males had a flu shot.
- 12 males did not have a flu shot.
- So, there are 51 males in total (39 + 12).
2. Identify the Question:
- We are asked to find the probability that a dormitory resident chosen at random has had a flu shot, given that he is male.
- This is conditional probability.
3. Define Conditional Probability:
- Conditional probability [tex]\( P(A|B) \)[/tex] is the probability of event [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has occurred.
- Here, [tex]\( A \)[/tex] is the event "having had a flu shot" and [tex]\( B \)[/tex] is the event "being male".
4. Calculate the Required Probability:
- We need to find [tex]\( P(\text{Had Flu Shot}|\text{Male}) \)[/tex].
- From the data, we know:
- The total number of males is 51.
- The number of males who had a flu shot is 39.
- The probability that a randomly chosen male has had a flu shot can be calculated as:
[tex]\[
P(\text{Had Flu Shot}|\text{Male}) = \frac{\text{Number of males who had a flu shot}}{\text{Total number of males}} = \frac{39}{51}
\][/tex]
5. Simplify the Fraction:
- Simplifying [tex]\(\frac{39}{51}\)[/tex] might give us a clearer fraction, but here we use the result known:
[tex]\[
\frac{39}{51} \approx 0.7647
\][/tex]
6. State the Final Answer:
- Therefore, the probability that a dormitory resident chosen at random has had a flu shot, given that he is male, is approximately 0.765 or 76.47%.
So, the probability is approximately [tex]\( 0.7647058823529411 \)[/tex].
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