Question

quincy is the proud owner of an eclectic bow tie collection. he keeps track of how many bow ties he has and organizes them by pattern and material. the probability that a bow tie is a solid color is 0.9, and the probability that it is made of velvet is 0.3. what is the probability that a randomly chosen bow tie is a solid color or is made of velvet if those are independent events? write your answer as a whole number, decimal, or simplified fraction.

Explanation:

Step1: Recall the formula for the probability of the union of two independent events.

The formula for \( P(A \cup B) \) when \( A \) and \( B \) are independent events is \( P(A \cup B)=P(A)+P(B)-P(A) \times P(B) \). Let \( A \) be the event that a bow tie is solid color (\( P(A) = 0.9 \)) and \( B \) be the event that a bow tie is made of velvet (\( P(B)=0.3 \)).

Step2: Substitute the values into the formula.

Substitute \( P(A) = 0.9 \), \( P(B)=0.3 \) into the formula:
\( P(A \cup B)=0.9 + 0.3-0.9\times0.3 \)
First, calculate the product \( 0.9\times0.3 = 0.27 \).
Then, calculate \( 0.9+0.3=1.2 \).
Finally, calculate \( 1.2 - 0.27=0.93 \).

Answer:

\( 0.93 \)