Question

which of the following are true if events a and b are independent? select all that apply.

a. ( p(a | b)=p(a) )
b. ( p(a | b)=p(b) )
c. ( p(a | b)=p(a \text { and } b) )
d. ( p(b | a)=p(a \text { and } b) )
e. ( p(b | a)=p(a) )
f. ( p(b | a)=p(b) )

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) (when \(P(B)>0\)) and \(P(B|A)=\frac{P(A\cap B)}{P(A)}\) (when \(P(A)>0\)).

Step2: Use the property of independent events

If \(A\) and \(B\) are independent, then \(P(A\cap B)=P(A)\times P(B)\).

• For \(P(A|B)\):

Substitute \(P(A\cap B) = P(A)\times P(B)\) into \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). We get \(P(A|B)=\frac{P(A)\times P(B)}{P(B)}=P(A)\) (since \(P(B)>0\)).

• For \(P(B|A)\):

Substitute \(P(A\cap B)=P(A)\times P(B)\) into \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). We get \(P(B|A)=\frac{P(A)\times P(B)}{P(A)}=P(B)\) (since \(P(A)>0\)).

Answer:

A. \(P(A|B) = P(A)\), F. \(P(B|A)=P(B)\)