Question

brian designs bow ties for a local boutique store. the store manager counted the ties currently on display, tracking them by material and shape. the probability that a bow tie is made of cotton is 0.4, and the probability that it is shaped like a batwing is 0.4. what is the probability that a randomly chosen bow tie is made of cotton or is shaped like a batwing 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 \( P(A \cup B)=P(A)+P(B)-P(A)P(B) \), where \( A \) is the event that a bow tie is made of cotton and \( B \) is the event that a bow tie is shaped like a batwing. We know that \( P(A) = 0.4 \) and \( P(B)=0.4 \).

Step2: Substitute the values into the formula.

First, calculate \( P(A)P(B) \): \( 0.4\times0.4 = 0.16 \).
Then, substitute into the formula for \( P(A \cup B) \): \( P(A \cup B)=0.4 + 0.4- 0.16 \).
Calculate the sum and difference: \( 0.4+0.4 = 0.8 \), and \( 0.8 - 0.16=0.64 \).

Answer:

\( 0.64 \)