Question

which two events are independent? a and x a and y b and x b and y

	x	y	z	total
b	5	8	7	20
c	30	15	5	50
total	50	28	22	100

Explanation:

Step1: Recall independence formula

Two events \(E\) and \(F\) are independent if \(P(E\cap F)=P(E)\times P(F)\). Let \(n(E\cap F)\) be the number of elements in the intersection of \(E\) and \(F\), \(n(E)\) be the number of elements in \(E\), \(n(F)\) be the number of elements in \(F\) and \(n(S) = 100\) (total number of elements). Then \(P(E\cap F)=\frac{n(E\cap F)}{n(S)}\), \(P(E)=\frac{n(E)}{n(S)}\) and \(P(F)=\frac{n(F)}{n(S)}\), and the independence condition becomes \(n(E\cap F)=\frac{n(E)\times n(F)}{n(S)}\).

Step2: Check A and X

\(n(A) = 30\), \(n(X)=50\), \(n(A\cap X)=15\). \(\frac{n(A)\times n(X)}{n(S)}=\frac{30\times50}{100}=15=n(A\cap X)\).

Step3: Check A and Y

\(n(A) = 30\), \(n(Y)=28\), \(\frac{n(A)\times n(Y)}{n(S)}=\frac{30\times28}{100}=8.4
eq n(A\cap Y) = 5\).

Step4: Check B and X

\(n(B)=20\), \(n(X)=50\), \(\frac{n(B)\times n(X)}{n(S)}=\frac{20\times50}{100}=10
eq n(B\cap X)=5\).

Step5: Check B and Y

\(n(B)=20\), \(n(Y)=28\), \(\frac{n(B)\times n(Y)}{n(S)}=\frac{20\times28}{100}=5.6
eq n(B\cap Y)=8\).

Answer:

A. A and X