Question

people at the state fair were surveyed about which type of lemonade they preferred. the results are shown below.
pink lemonade: 156 males, 72 females
yellow lemonade: 104 males, 48 females
the events \prefers pink lemonade\ and \female\ are independent because
\\( p(\text{pink lemonade} | \text{female}) = p(\text{pink lemonade}) = 0.6, \\)
\\( p(\text{female} | \text{pink lemonade} ) = p(\text{pink lemonade}) = 0.3, \\)
\\( p(\text{pink lemonade} | \text{female}) = 0.3 \\) and \\( p(\text{pink lemonade}) = 0.6, \\)
\\( p(\text{female} | \text{pink lemonade} ) = 0.3 \\) and \\( p(\text{pink lemonade}) = 0.6. \\)

Explanation:

Step1: Calculate total survey respondents

Total = 156 + 72 + 104 + 48 = 380

Step2: Calculate P(pink lemonade)

Total pink lemonade = 156 + 72 = 228
$P(\text{pink lemonade}) = \frac{228}{380} = 0.6$

Step3: Calculate total female respondents

Total females = 72 + 48 = 120

Step4: Calculate P(pink lemonade | female)

$P(\text{pink lemonade} | \text{female}) = \frac{72}{120} = 0.6$

Step5: Verify independence rule

Two events A and B are independent if $P(A|B) = P(A)$. Here, $P(\text{pink lemonade} | \text{female}) = P(\text{pink lemonade}) = 0.6$, so they are independent.

Answer:

A. $P(\text{pink lemonade} | \text{female}) = P(\text{pink lemonade}) = 0.6$