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(pink lemonade | female) = p(pink lemonade) = 0.6. p(female | pink lemonade ) = p(pink lemonade) = 0.3. p(pink lemonade | female) = 0.3 and p(pink lemonade) = 0.6. p(female | pink lemonade ) = 0.3 and p(pink lemonade) = 0.6.

Explanation:

Step1: Calculate total number of people

Total number of people = \(156 + 72+104 + 48=380\).

Step2: Calculate \(P(\text{pink lemonade})\)

Number of people who prefer pink lemonade = \(156 + 72 = 228\). So \(P(\text{pink lemonade})=\frac{228}{380}=0.6\).

Step3: Calculate number of females

Number of females = \(72 + 48=120\).

Step4: Calculate \(P(\text{pink lemonade}|\text{female})\)

Number of females who prefer pink lemonade = 72. So \(P(\text{pink lemonade}|\text{female})=\frac{72}{120}=0.6\).

Step5: Check independence condition

Two events \(A\) and \(B\) are independent if \(P(A|B)=P(A)\). Here \(A\) is the event of preferring pink - lemonade and \(B\) is the event of being female. Since \(P(\text{pink lemonade}|\text{female}) = P(\text{pink lemonade})=0.6\), the events are independent.

Answer:

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