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For two programs at a university, the type of student for two majors is as follows:

\begin{tabular}{|c|c|c|c|}
\hline & History & Science & Total \\
\hline Undergraduate & 390 & 422 & 812 \\
\hline Graduate & 73 & 188 & 261 \\
\hline Total & 463 & 610 & 1073 \\
\hline
\end{tabular}

Find the probability that a student is a graduate student, given they are a history major.

[tex]\[ P(\text{graduate} \mid \text{history}) = \frac{P(\text{graduate and history})}{P(\text{history})} = [?] \][/tex]

Round to the nearest hundredth.

Sagot :

GinnyAnswer
To find the probability that a student is a graduate student given that they are a history major, we can use conditional probability. The formula for conditional probability is given by:

[tex]\[ P(\text{graduate} \mid \text{history}) = \frac{P(\text{graduate and history})}{P(\text{history})} \][/tex]

From the given table:

- The number of students who are history majors (i.e., [tex]\( \text{Total history} \)[/tex]) is 463.
- The number of students who are both graduate students and history majors (i.e., [tex]\( \text{Graduate history} \)[/tex]) is 73.

We need to find the probability [tex]\( P(\text{graduate} \mid \text{history}) \)[/tex]:

[tex]\[ P(\text{graduate} \mid \text{history}) = \frac{\text{Graduate history}}{\text{Total history}} \][/tex]
[tex]\[ P(\text{graduate} \mid \text{history}) = \frac{73}{463} \][/tex]

Now, perform the division to get the probability:

[tex]\[ P(\text{graduate} \mid \text{history}) = \frac{73}{463} \approx 0.15766738660907129 \][/tex]

To round this to the nearest hundredth:

[tex]\[ P(\text{graduate} \mid \text{history}) \approx 0.16 \][/tex]

Therefore, the probability that a student is a graduate student given that they are a history major is approximately [tex]\( 0.16 \)[/tex].
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