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At West View High School, every freshman (Fr) and sophomore (So) has either math (M), science (S), English (E), or history (H) as the first class of the day. The two-way table shows the distribution of students by first class and grade level.

\begin{tabular}{|c|c|c|c|c|c|}
\cline{2-6}
\multicolumn{1}{c|}{} & M & S & E & H & Total \\
\hline
Fr & 78 & 32 & 59 & 43 & 212 \\
\hline
So & 38 & 65 & 42 & 51 & 196 \\
\hline
Total & 116 & 97 & 101 & 94 & 408 \\
\hline
\end{tabular}

Which expression represents the conditional probability that a randomly selected freshman has English as the first class of the day?

A. [tex]\( P(E \mid Fr) \)[/tex]

B. [tex]\( P(Fr \mid E) \)[/tex]



Answer :

GinnyAnswer
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.

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