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

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

Explanation:

Definition: Two events are independent if

`P(A)\cdot P(B)=P(B\cap A)\\$Equivalently:$\\P(B)=(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)=(P(B\cap A))/(P(A))`

Left Hand Side =1/6

Right Hand Side =(1/6)÷(1/2)=1/3

Since
`P(B)\\eq (P(B\cap A))/(P(A))`, therefore
`P(B)\\eq P(B|A)` which makes the events dependent.