Question

a college finds that 10% of students have taken a distance learning class and that 40% of students are part - time students. of the part - time students, 20% have taken a distance learning class. let d = event that a student takes a distance learning class and e = event that a student is a part - time student. part (a) part (b) part (c) find p(d or e) part (d) part (e)

Explanation:

Step1: Recall the formula for \( P(D \text{ OR } E) \)

The formula for the probability of the union of two events \( D \) and \( E \) is \( P(D \cup E)=P(D)+P(E)-P(D \cap E) \).

Step2: Identify given probabilities

We know:

• \( P(D) = 0.10 \) (10% of students have taken a distance learning class),
• \( P(E) = 0.40 \) (40% of students are part - time students),
• \( P(D|E)=0.20 \) (20% of part - time students have taken a distance learning class).

Step3: Calculate \( P(D \cap E) \)

Using the formula for conditional probability \( P(D|E)=\frac{P(D \cap E)}{P(E)} \), we can solve for \( P(D \cap E) \).
Rearranging the formula gives \( P(D \cap E)=P(D|E)\times P(E) \).
Substituting the known values: \( P(D \cap E)=0.20\times0.40 = 0.08 \).

Step4: Calculate \( P(D \cup E) \)

Substitute \( P(D) = 0.10 \), \( P(E)=0.40 \), and \( P(D \cap E) = 0.08 \) into the formula \( P(D \cup E)=P(D)+P(E)-P(D \cap E) \).
\( P(D \cup E)=0.10 + 0.40-0.08=0.42 \).

Answer:

\( 0.42 \)