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.

M S E H Total

Fr 78 32 59 43 212

So 38 65 42 51 196

Total 116 97 101 94 408

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

$P($ □ )

What is the probability that a randomly selected freshman has English as the first class of the day?

□

Question

01-11-2026
Mathematics

Answered

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.

M S E H Total

Fr 78 32 59 43 212

So 38 65 42 51 196

Total 116 97 101 94 408

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

$P($ □ )

What is the probability that a randomly selected freshman has English as the first class of the day?

□

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T15:25:08-05:00

The event is selecting an English student ($E$) given the student is a freshman ($\text{Fr}$).
Using the formula $P(E|\text{Fr}) = \frac{n(E \cap \text{Fr})}{n(\text{Fr})}$, we substitute the values from the table: $n(E \cap \text{Fr}) = 59$ and $n(\text{Fr}) = 212$.
The expression for the probability is $P(E|\text{Fr})$.
The final calculated probability is $\boxed{\frac{59}{212}}$.

Explanation

Analyze the problem and identify data from the table

We are given a two-way table representing the number of freshmen and sophomores at West View High School based on their first class of the day (Math, Science, English, or History). We need to find the conditional probability that a randomly selected student is taking English, given that the student is a freshman.

The total number of freshmen is given in the table row labeled "Fr" under the "Total" column, which is $n(\text{Fr}) = 212$. The number of freshmen who have English as their first class is given in the intersection of the "Fr" row and the "E" column, which is $n(E \cap \text{Fr}) = 59$.

Determine the conditional probability expression and value

The conditional probability $P(E|\text{Fr})$ is the probability that a student is taking English given that they are a freshman. By definition, this is:
\[P(E|\text{Fr}) = \frac{n(E \cap \text{Fr})}{n(\text{Fr})}\]
where $n(E \cap \text{Fr})$ is the number of freshmen taking English, and $n(\text{Fr})$ is the total number of freshmen. Substituting the values from the table:
\[P(E|\text{Fr}) = \frac{59}{212}\]

Conclusion and final result

The expression for the conditional probability is $P(E|\text{Fr})$. The numerical value is $\frac{59}{212}$. This fraction cannot be simplified further, as 59 is a prime number and is not a factor of 212.

Examples

Conditional probability is frequently used in school administration and data analysis to understand student demographics. For example, if a school counselor wants to know the likelihood that a freshman student has a specific class to ensure balanced scheduling, they use the row of total freshmen as their sample space to calculate proportions like $\frac{59}{212} \approx 0.278$, or roughly 27.8%.