To find the probability that a student is a science major given that they are a graduate student, denoted as [tex]\( P(\text{science} \mid \text{graduate}) \)[/tex], we use the formula:
[tex]\[
P(\text{science} \mid \text{graduate}) = \frac{P(\text{science and graduate})}{P(\text{graduate})}
\][/tex]
Here's a step-by-step explanation:
1. Determine [tex]\( P(\text{science and graduate}) \)[/tex]:
- The number of students who are both science majors and graduate students is given in the table as 188.
- Since this is the exact count and we are dealing with probabilities, it directly reflects the probability when divided by the total number of students overall, but for our specific case, it will be used to find a conditional probability.
2. Determine [tex]\( P(\text{graduate}) \)[/tex]:
- The total number of graduate students is given in the table as 261.
3. Calculate [tex]\( P(\text{science} \mid \text{graduate}) \)[/tex]:
- We divide the number of science graduates by the total number of graduates:
[tex]\[
P(\text{science} \mid \text{graduate}) = \frac{188}{261}
\][/tex]
4. Perform the Division:
- Compute the value:
[tex]\[
\frac{188}{261} \approx 0.7203
\][/tex]
5. Round to the Nearest Hundredth:
- To round 0.7203 to the nearest hundredth, we look at the thousandths place. Since it is '0', we round down.
[tex]\[
0.7203 \approx 0.72
\][/tex]
Therefore, the probability that a student is a science major given that they are a graduate student is approximately [tex]\( 0.72 \)[/tex] when rounded to the nearest hundredth.
So,
[tex]\[ P(\text{science} \mid \text{graduate}) \approx 0.72 \][/tex]
