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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 :

Answer:

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

Step-by-step explanation:

Definition: Two events are independent if

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

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

  • [tex]P(A)=3/6=1/2[/tex]

Event B is "rolling a 2."

  • P(B)=1/6

[tex]\{A\cap B\}={2}\\P(A\cap B)=1/6[/tex]

Substitution into [tex]P(B)=\frac{P(B\cap A)}{P(A)}[/tex]

Left Hand Side =1/6

Right Hand Side =(1/6)÷(1/2)=1/3

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

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