Questions: Sam's closet contains blue and green shirts. He has eight blue shirts, and seven green shirts. Five of the blue shirts have stripes, four of the green shirts have stripes. What is the probability that Sam randomly chooses a shirt that is blue or has stripes? A. P(B ∪ S)=0.33 B. P(B ∪ S)=0.69 C. P(B ∪ S)=0.27 D. P(B ∪ S)=0.8

Sam's closet contains blue and green shirts. He has eight blue shirts, and seven green shirts. Five of the blue shirts have stripes, four of the green shirts have stripes. What is the probability that Sam randomly chooses a shirt that is blue or has stripes?
A. P(B ∪ S)=0.33
B. P(B ∪ S)=0.69
C. P(B ∪ S)=0.27
D. P(B ∪ S)=0.8

Transcript text: Sam's closet contains blue and green shirts. He has eight blue shirts, and seven green shirts. Five of the blue shirts have stripes, four of the green shirts have stripes. What is the probability that Sam randomly chooses a shirt that is blue or has stripes? A. $P(B \cup S)=0.33$ B. $P(B \cup S)=0.69$ C. $P(B \cup S)=0.27$ D. $P(B \cup S)=0.8$

Solution

Solution Steps

To find the probability that Sam randomly chooses a shirt that is blue or has stripes, we need to use the principle of inclusion and exclusion.

Calculate the total number of shirts.
Calculate the number of blue shirts.
Calculate the number of shirts with stripes.
Calculate the number of blue shirts with stripes.
Use the formula for the union of two events: $ P(B \cup S) = P(B) + P(S) - P(B \cap S) $.

Step 1: Calculate the Total Number of Shirts

The total number of shirts is the sum of blue and green shirts: \[ \text{Total shirts} = 8 + 7 = 15 \]

Step 2: Calculate the Probability of Choosing a Blue Shirt

The probability of choosing a blue shirt ($P(B)$) is the number of blue shirts divided by the total number of shirts: \[ P(B) = \frac{8}{15} \approx 0.5333 \]

Step 3: Calculate the Probability of Choosing a Striped Shirt

The probability of choosing a striped shirt ($P(S)$) is the number of striped shirts divided by the total number of shirts: \[ P(S) = \frac{9}{15} = 0.6 \]

Step 4: Calculate the Probability of Choosing a Blue Shirt with Stripes

The probability of choosing a blue shirt with stripes ($P(B \cap S)$) is the number of blue shirts with stripes divided by the total number of shirts: \[ P(B \cap S) = \frac{5}{15} \approx 0.3333 \]

Step 5: Calculate the Probability of Choosing a Shirt that is Blue or has Stripes

Using the principle of inclusion and exclusion, the probability of choosing a shirt that is blue or has stripes ($P(B \cup S)$) is: \[ P(B \cup S) = P(B) + P(S) - P(B \cap S) \] Substituting the values: \[ P(B \cup S) = 0.5333 + 0.6 - 0.3333 = 0.8 \]