Question

Murphy’s math teacher sometimes wears scarves to class. Murphy has been documenting the relationship between his teacher wearing a scarf and when the class has a math quiz. The probabilities are as follows:


• P(wearing a scarf) = 10%


• P(math quiz) = 15%


• P(wearing a scarf and math quiz) = 5%


Are the events “the teacher is wearing a scarf” and “there will be a quiz” independent events? Explain.

Answer 1

Answer:

See below ~

Step-by-step explanation:

Probability Equation

• P (A ∪ B) = P (A) + P (B) - P (A ∩ B)

Let :

• P (wearing a scarf) = P (A)
• P (math quiz) = P (B)

Solving the equation

• 5% = 10% + 15% - P (A ∩ B)
• 5% = 25% - P (A ∩ B)
• P (A ∩ B) = 20%

For independent events, P (A ∩ B) = 0% = ∅

Therefore, they are not independent events.

Answer 2

Answer:

Let "wearing a scarf" = S

Let "math quiz" = Q

Given:

• P(S) = 10% = 0.1
• P(Q) = 15% = 0.15
• P(S ∩ Q) = 5% = 0.05

The product law for independent events A and B is:

P(A ∩ B) = P(A)P(B)

Therefore, to determine if events S and Q are independent:

⇒ P(S)P(Q) = 0.1 × 0.15 = 0.015 = 1.5%

Therefore, as 1.5% ≠ 5% the events are not independent.