which statement is true about whether c and y are independent events? c and y are independent events because…

which statement is true about whether c and y are independent events? c and y are independent events because p(c|y)=p(y). c and y are independent events because p(c|y)=p(c). c and y are not independent events because p(c|y)≠p(y). c and y are not independent events because p(c|y)≠p(c).

which statement is true about whether c and y are independent events? c and y are independent events because p(c|y)=p(y). c and y are independent events because p(c|y)=p(c). c and y are not independent events because p(c|y)≠p(y). c and y are not independent events because p(c|y)≠p(c).

Answer

Answer:

D. C and Y are not independent events because (P(C|Y)\neq P(C)).

Explanation:

Step1: Calculate (P(C))

The total number of outcomes is (n = 300), and the number of outcomes in event (C) is (n_C=110). So (P(C)=\frac{n_C}{n}=\frac{110}{300}=\frac{11}{30}).

Step2: Calculate (P(C|Y))

The number of outcomes in event (Y) is (n_Y = 75), and the number of outcomes in both (C) and (Y) is (n_{C\cap Y}=35). By the formula for conditional - probability (P(C|Y)=\frac{P(C\cap Y)}{P(Y)}), and since (P(C\cap Y)=\frac{n_{C\cap Y}}{n}=\frac{35}{300}), (P(Y)=\frac{n_Y}{n}=\frac{75}{300}), then (P(C|Y)=\frac{n_{C\cap Y}}{n_Y}=\frac{35}{75}=\frac{7}{15}).

Step3: Compare (P(C)) and (P(C|Y))

(\frac{11}{30}\neq\frac{7}{15}) (since (\frac{7}{15}=\frac{14}{30})), so (P(C|Y)\neq P(C)). Two events (C) and (Y) are independent if and only if (P(C|Y) = P(C)). Since (P(C|Y)\neq P(C)), (C) and (Y) are not independent events.