Use the table to answer the question.

Question

Color
Shape Red Yellow TOTAL
Circle 12 18 30
Triangle 20 30 50
Total 32 48 80
Let event A
represent the P(yellow)
and let event B
represent the P(yellow|triangle)
. Are they independent? Explain.

(1 point)
Responses

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 .

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 .

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 check if P(A|B) = P(A).

1. Calculating P(A):

   • P(A) is the probability of selecting a yellow shape from all shapes.
   • There are 48 yellow shapes out of a total of 80 shapes.
   • Thus, P(A) = 48/80 = 0.6.

2. Calculating P(B):

   • P(B) is the probability of selecting a yellow shape given that the shape is a triangle.
   • There are 30 yellow triangles out of a total of 50 triangles.
   • Thus, P(B) = 30/50 = 0.6.

3. Calculating P(A|B):

   • P(A|B) is the probability of selecting a yellow shape given that we have a triangle.
   • Since we are already considering only triangles, and out of 50 triangles, we have 30 that are yellow, P(A|B) = 30/50 = 0.6.

Since P(A) = 0.6 and P(A|B) = 0.6, we can see that:

P(A|B) = P(A)

This indicates that events A and B are independent.

The correct response is:

• Yes, because P(A|B) = P(A).