Question

PLEASE HELP ME OUT :) Independent or Dependent?

Also, is there an easier way to determine if these are independent or dependent other than working out the entire problem? (Not trying to be lazy. My school website uses common core learning which explains each problem in, seemingly, the most difficult way possible for most different types of algebra and geometry)

Answer 1

Option 1.

`P(A) = (1)/(8),\ P(A|B) = (1)/(3)` Dependent

Option 2

`P(A) = (1)/(4),\ P(A|B) = (1)/(4)` Independent

Option 3

`P(B) = (1)/(8),\ P(B|A) = (1)/(4)` Dependent

Option 4

`P(B) = (1)/(4),\ P(B|A) = (1)/(4)` Independent

Explanation:

Two events A and B are independent if the occurrence of A does not affect the probability of B.

On the other hand The probability of A given B is defined as:

`P (A | B) = (P (A\ and\ B))/(P (B))`

When two events are independent then:

`P (A\ and\ B) = P (A) * P (B)`

So if the two events A and B are independent this means that:

`P (A | B) = (P (A) * P (B))/(P (B))`

`P (A | B) = P (A)`

Which makes sense because if the events are independent then the probability of A not being affected by B.

So to solve this problem identify in what cases

`P (A | B) = P (A)` or
`P (B | A) = P (B)`

When this happens those events are independent

Option 1.

`P(A) = (1)/(8),\ P(A|B) = (1)/(3)` Dependent

Option 2

`P(A) = (1)/(4),\ P(A|B) = (1)/(4)` Independent

Option 3

`P(B) = (1)/(8),\ P(B|A) = (1)/(4)` Dependent

Option 4

`P(B) = (1)/(4),\ P(B|A) = (1)/(4)` Independent