Question

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).

Explanation:

Step1: Recall the definition of independent events

Two events \(C\) and \(Y\) are independent if \(P(C|Y)=P(C)\). The formula for conditional - probability is \(P(C|Y)=\frac{P(C\cap Y)}{P(Y)}\), and \(P(C)=\frac{n(C)}{n(S)}\), where \(n(C)\) is the number of elements in event \(C\), \(n(Y)\) is the number of elements in event \(Y\), and \(n(S)\) is the total number of elements in the sample - space.

Step2: Calculate \(P(C)\)

The total number of elements in the sample - space \(n(S) = 300\), and the number of elements in event \(C\) is \(n(C)=110\). So, \(P(C)=\frac{110}{300}=\frac{11}{30}\).

Step3: Calculate \(P(C\cap Y)\)

From the table, the number of elements in \(C\cap Y\) is \(35\). So, \(P(C\cap Y)=\frac{35}{300}=\frac{7}{60}\).

Step4: Calculate \(P(Y)\)

The number of elements in event \(Y\) is \(n(Y)=75\). So, \(P(Y)=\frac{75}{300}=\frac{1}{4}\).

Step5: Calculate \(P(C|Y)\)

Using the formula \(P(C|Y)=\frac{P(C\cap Y)}{P(Y)}\), we substitute \(P(C\cap Y)=\frac{7}{60}\) and \(P(Y)=\frac{1}{4}\). Then \(P(C|Y)=\frac{\frac{7}{60}}{\frac{1}{4}}=\frac{7}{60}\times4=\frac{7}{15}\).
Since \(\frac{7}{15}
eq\frac{11}{30}\), i.e., \(P(C|Y)
eq P(C)\).

Answer:

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