Use the table to answer the question.

Question

Circle and Triangle Categorized into Different Colors
Color
Shape Red Yellow TOTAL
Circle 12 18 30
Triangle 20 30 50
Total 32 48 80
If event A is that a shape is a triangle and event B is that a shape is yellow, are the two events independent? Explain.

(1 point)
Responses

Yes, because P(A)=P(B).
Yes, because upper P left parenthesis upper A right parenthesis equals upper P left parenthesis upper B right parenthesis .

Yes, because P(A|B)=P(A).
Yes, because upper P left parenthesis upper A vertical bar upper B right parenthesis equals upper P left parenthesis upper A right parenthesis .

No, because both events do not include yellow.
No, because both events do not include yellow.

No, because both events do not include triangles.

Answer 1

To determine if events A and B are independent, we need to evaluate the relationship between P(A), P(B), and P(A|B), where:

• Event A = the shape is a triangle.
• Event B = the shape is yellow.

Step 1: Calculate P(A)

• P(A) = Probability of choosing a triangle
  = Number of triangles / Total shapes
  = 50 / 80
  = 0.625

Step 2: Calculate P(B)

• P(B) = Probability of choosing a yellow shape
  = Number of yellow shapes / Total shapes
  = 48 / 80
  = 0.6

Step 3: Calculate P(A ∩ B)

• P(A ∩ B) = Probability of choosing a yellow triangle
  = Number of yellow triangles / Total shapes
  = 30 / 80
  = 0.375

Step 4: Calculate P(A|B)

• P(A|B) = Probability of A given B
  = P(A ∩ B) / P(B)
  = (30 / 80) / (48 / 80)
  = 30 / 48
  = 0.625

Step 5: Check for Independence For events A and B to be independent, the following must be true:

• P(A|B) = P(A)

From our calculations:

• P(A) = 0.625
• P(A|B) = 0.625

Since P(A|B) = P(A), the events are independent.

Therefore, the correct response is:

Yes, because P(A|B) = P(A).