Question

this question has two parts. use the information to answer part a and part b. let e be the event that tom has english homework, and let h be the event that tom has history homework. p(e) = 0.6, p(h) = 0.4, p(e and h) = 0.3. part a: are e and h dependent or independent events? \bigcirc a. dependent \bigcirc b. independent. part b: which equation justifies why the events are dependent or independent? \bigcirc a. p(e \text{ and } h) \
eq p(e) + p(h) \bigcirc b. p(e \text{ and } h) \
eq p(e) \times p(h) \bigcirc c. p(e \text{ and } h) = p(e) - p(h) \bigcirc d. p(e \text{ and } h) = p(e) \times p(h)

Explanation:

Part A

Step1: Recall independence rule

For independent events, \( P(E \text{ and } H) = P(E) \times P(H) \).

Step2: Calculate \( P(E) \times P(H) \)

\( P(E) = 0.6 \), \( P(H) = 0.4 \), so \( P(E) \times P(H) = 0.6 \times 0.4 = 0.24 \).

Step3: Compare with \( P(E \text{ and } H) \)

Given \( P(E \text{ and } H) = 0.3 \), and \( 0.3
eq 0.24 \), so events are dependent.

Step1: Recall independence condition

Two events \( E \) and \( H \) are independent if \( P(E \text{ and } H) = P(E) \times P(H) \), dependent if not.

Step2: Check each option

• Option A: \( P(E)+P(H)=0.6 + 0.4 = 1.0 \), \( P(E \text{ and } H)=0.3 \), but addition rule is for mutually exclusive, not independence.
• Option B: \( P(E) \times P(H)=0.24 \), \( P(E \text{ and } H)=0.3 \), so \( P(E \text{ and } H)

eq P(E) \times P(H) \), which shows dependence.

• Option C: \( P(E)-P(H)=0.2

eq 0.3 \), invalid.

• Option D: \( P(E) \times P(H)=0.24

eq 0.3 \), so this is not true.

Answer:

A. dependent

Part B