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].
