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Answer :
To find the probability that a dormitory resident has had a flu shot given that he is male, we use the concept of conditional probability. The formula for conditional probability is
[tex]$$
P(\text{Had Flu Shot} \mid \text{Male}) = \frac{P(\text{Had Flu Shot and Male})}{P(\text{Male})}.
$$[/tex]
From the data provided:
- The number of male residents who had a flu shot is [tex]$39$[/tex].
- The total number of male residents is [tex]$51$[/tex].
Thus, the conditional probability is
[tex]$$
P(\text{Had Flu Shot} \mid \text{Male}) = \frac{39}{51}.
$$[/tex]
To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor. The fraction [tex]$\frac{39}{51}$[/tex] simplifies to
[tex]$$
\frac{39 \div 3}{51 \div 3} = \frac{13}{17}.
$$[/tex]
Therefore, the probability that a dormitory resident chosen at random has had a flu shot, given that he is male, is
[tex]$$
\boxed{\frac{13}{17}}.
$$[/tex]
[tex]$$
P(\text{Had Flu Shot} \mid \text{Male}) = \frac{P(\text{Had Flu Shot and Male})}{P(\text{Male})}.
$$[/tex]
From the data provided:
- The number of male residents who had a flu shot is [tex]$39$[/tex].
- The total number of male residents is [tex]$51$[/tex].
Thus, the conditional probability is
[tex]$$
P(\text{Had Flu Shot} \mid \text{Male}) = \frac{39}{51}.
$$[/tex]
To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor. The fraction [tex]$\frac{39}{51}$[/tex] simplifies to
[tex]$$
\frac{39 \div 3}{51 \div 3} = \frac{13}{17}.
$$[/tex]
Therefore, the probability that a dormitory resident chosen at random has had a flu shot, given that he is male, is
[tex]$$
\boxed{\frac{13}{17}}.
$$[/tex]
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