Question

james surveyed people at school and asked whether they bring their lunch to school or buy their lunch at school more often. the results are shown below. bring lunch: 46 males, 254 females. buy lunch: 176 males, 264 females. the events male and buys lunch are not independent because p(buys lunch | male) = p(male) = 0.4 p(male | buys lunch) = p(male) = 0.3 p(buys lunch | male) = 0.3 and p(male) = 0.4 p(male | buys lunch) = 0.4 and p(male) = 0.3

Explanation:

Step1: Calculate total number of people

Total number of people = \(46 + 254+176 + 264=740\)

Step2: Calculate \(P(\text{male})\)

Number of males = \(46 + 176=222\), so \(P(\text{male})=\frac{222}{740}=0.3\)

Step3: Calculate \(P(\text{buys lunch})\)

Number of people who buy lunch = \(176 + 264 = 440\), so \(P(\text{buys lunch})=\frac{440}{740}\)

Step4: Calculate \(P(\text{male and buys lunch})\)

Number of males who buy lunch = \(176\), so \(P(\text{male and buys lunch})=\frac{176}{740}\)

Step5: Calculate \(P(\text{male}|\text{buys lunch})\)

By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), \(P(\text{male}|\text{buys lunch})=\frac{\frac{176}{740}}{\frac{440}{740}}=\frac{176}{440}=0.4\)

Since \(P(\text{male}|\text{buys lunch}) = 0.4\) and \(P(\text{male})=0.3\), the events "male" and "buys lunch" are not independent.

Answer:

\(P(\text{male}|\text{buys lunch}) = 0.4\) and \(P(\text{male}) = 0.3\)