james surveyed people at school and asked whether they bring their lunch to school or buy their lunch at…

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\n○ p(buys lunch | male) = p(male) = 0.4.\n○ p(male | buys lunch) = p(male) = 0.3.\n○ p(buys lunch | male) = 0.3 and p(male) = 0.4.\n○ p(male | buys lunch) = 0.4 and p(male) = 0.3.

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\n○ p(buys lunch | male) = p(male) = 0.4.\n○ p(male | buys lunch) = p(male) = 0.3.\n○ p(buys lunch | male) = 0.3 and p(male) = 0.4.\n○ p(male | buys lunch) = 0.4 and p(male) = 0.3.

Answer

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{buys lunch}|\text{male}))

By the formula (P(A|B)=\frac{P(A\cap B)}{P(B)}), (P(\text{buys lunch}|\text{male})=\frac{P(\text{buys lunch and male})}{P(\text{male})}=\frac{\frac{176}{740}}{\frac{222}{740}}=\frac{176}{222}\approx0.79\neq0.3).

Step6: 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{P(\text{male and buys lunch})}{P(\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:

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