Question

Use the table to answer the 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 No, because both events do not include yellow. No, because both events do not include yellow. 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 triangles. No, because both events do not include triangles. 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 .

Answer 1

To determine if events A (a shape is a triangle) and B (a shape is yellow) are independent, we need to check if \( P(A|B) = P(A) \).

We can calculate each probability using the provided data from the table:

1. Total outcomes = 80 (given).

2. Probability of event A (P(A)):

   • Number of triangles = 50.
   • \( P(A) = \frac{\text{Number of triangles}}{\text{Total outcomes}} = \frac{50}{80} = 0.625 \).

3. Probability of event B (P(B)):

   • Number of yellow shapes = 48.
   • \( P(B) = \frac{\text{Number of yellow shapes}}{\text{Total outcomes}} = \frac{48}{80} = 0.6 \).

4. Probability of event A given B (P(A|B)):

   • Number of yellow triangles = 30 (from the table).
   • \( P(A|B) = \frac{\text{Number of yellow triangles}}{\text{Total yellow shapes}} = \frac{30}{48} = 0.625 \).

Now, we can compare \( P(A|B) \) and \( P(A) \):

• \( P(A|B) = 0.625 \)
• \( P(A) = 0.625 \)

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

Thus, the correct response is:

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