Question

An experiment consists of rolling a standard six-sided die once. Event A is "rolling an even number" and event B is "rolling a 2."
Are the events dependent or independent? Why?


Select the option that correctly answers both questions.

Events A and B are independent, because P(B)=P(B|A)=1/6.

Events A and B are dependent, because P(A)≠P(B|A).

Events A and B are independent, because P(A)=P(B|A)=1/2.

Events A and B are dependent, because P(B)≠P(B|A).

Answer 1

Answer:

(D)Events A and B are dependent, because P(B)≠P(B|A).

Step-by-step explanation:

Definition: Two events are independent if

$P(A)\cdot P(B)=P(B\cap A)\\ Equivalently: \\P(B)=\frac{P(B\cap A)}{P(A)} \\P(B)=P(B|A)$

An experiment consists of rolling a standard six-sided die once.

Event A is "rolling an even number"

Even numbers are 2,4 and 6

• $P(A)=3/6=1/2$

Event B is "rolling a 2."

• P(B)=1/6

$\{A\cap B\}={2}\\P(A\cap B)=1/6$

Substitution into $P(B)=\frac{P(B\cap A)}{P(A)}$

Left Hand Side =1/6

Right Hand Side =(1/6)÷(1/2)=1/3

Since $P(B)\neq \frac{P(B\cap A)}{P(A)}$, therefore $P(B)\neq P(B|A)$ which makes the events dependent.