To find the conditional probability that a randomly selected freshman has English as the first class of the day, we need to understand this probability in the form of [tex]\( P(E|Fr) \)[/tex]. This represents the probability that a student has English ([tex]\(E\)[/tex]) given that they are a freshman ([tex]\(Fr\)[/tex]).
### Step-by-Step Solution:
1. Identify Relevant Information:
- The total number of freshmen: 212
- The number of freshmen who have English as their first class: 59
2. Definition of Conditional Probability:
The conditional probability [tex]\( P(E|Fr) \)[/tex] is defined as:
[tex]\[
P(E|Fr) = \frac{P(E \cap Fr)}{P(Fr)}
\][/tex]
Here, [tex]\( P(E \cap Fr) \)[/tex] is the probability that a student has English as their first class and is a freshman. [tex]\( P(Fr) \)[/tex] is the probability that a student is a freshman.
3. Calculate Probabilities:
- Total freshmen ( [tex]\( P(Fr) \)[/tex] ) = 212
- Freshmen with English as first class ( [tex]\( P(E \cap Fr) \)[/tex] ) = 59
4. Conditional Probability Calculation:
[tex]\[
P(E|Fr) = \frac{\text{Number of freshmen with English}}{\text{Total number of freshmen}} = \frac{59}{212}
\][/tex]
5. Final Value:
By calculating the value, we get:
[tex]\[
P(E|Fr) \approx 0.2783018867924528
\][/tex]
Therefore, the conditional probability that a randomly selected freshman has English as the first class of the day is [tex]\(\boxed{0.2783018867924528}\)[/tex].
In terms of the given expressions:
- [tex]\( P(E|Fr) \)[/tex] correctly represents the conditional probability sought.
- The numerical value computed represents this conditional probability.
