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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 dormitories to see whether they received a preventative flu shot. The results are shown below.

What is the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male?

Residents at College Dormitories:

\[
\begin{array}{|c|c|c|}
\hline
& \text{Male} & \text{Female} \\
\hline
\text{Had Flu Shot} & 39 & 41 \\
\hline
\begin{array}{c}
\text{Didn't Have} \\
\text{Flu Shot}
\end{array} & 12 & 8 \\
\hline
\text{Total} & 51 & 49 \\
\hline
\end{array}
\]

A. [tex] \frac{13}{17} [/tex]
B. [tex] \frac{51}{100} [/tex]
C. [tex] \frac{39}{100} [/tex]
D. [tex] \frac{39}{80} [/tex]

Answer :

To determine the probability that a randomly chosen male dormitory resident has had a flu shot, we start by analyzing the data from the given table.

Step-by-step Calculation


  1. Identify the Total Number of Male Residents:


    • According to the table, there are a total of [tex]51[/tex] male residents.



  2. Identify the Number of Male Residents Who Had a Flu Shot:


    • The table shows that [tex]39[/tex] male residents received a flu shot.



  3. Calculate the Probability:


    • The probability [tex]P[/tex] that a randomly selected male resident has had a flu shot is given by the fraction of males who received the shot out of the total number of males:


    [tex]P(\text{Flu Shot | Male}) = \frac{39}{51}[/tex]


  4. Simplify the Fraction:


    • Simplifying [tex]\frac{39}{51}[/tex] by finding the greatest common divisor (GCD) of [tex]39[/tex] and [tex]51[/tex], which is [tex]3[/tex], gives:


    [tex]\frac{39 \div 3}{51 \div 3} = \frac{13}{17}[/tex]



Thus, the probability that a randomly chosen male resident has had a flu shot is [tex]\frac{13}{17}[/tex].

Conclusion

Therefore, the correct answer is Option A: [tex]\frac{13}{17}[/tex].

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Rewritten by : Barada

To find the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, we will use the concept of conditional probability.

Step-by-step Approach:


  1. Identify the Relevant Information: We need to find the probability that a male resident has had a flu shot. From the table, there are 51 male residents in total.


  2. Determine the Number of Males Who Had a Flu Shot: The table indicates that 39 males had a flu shot.


  3. Use the Conditional Probability Formula: The probability of an event A given event B is given by:

    [tex]P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]

    Here, A is the event of having had a flu shot, and B is the event of being male.


  4. Apply the Formula:

    [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 values, we get:

    [tex]P(\text{Had Flu Shot} | \text{Male}) = \frac{39}{51}[/tex]


  5. Simplify the Fraction: To fully simplify, we divide both the numerator and the denominator by 3:

    [tex]\frac{39}{51} = \frac{13}{17}[/tex]



Thus, the probability that a dormitory resident chosen at random from this group has had a flu shot, given that he is male, is [tex]\frac{13}{17}[/tex].

Therefore, the correct answer is A. [tex]\frac{13}{17}[/tex].