Question

Consider 6 different-color marbles that are randomly selected. Create a scenario for event A
and a scenario for event B
, such that P(A and B)=1/30
. State the result of event A
, the result of event B
, and explain if the two events are independent or dependent.(4 points)

Answer 1

Take two marbles drawn one after the other without replacement (order matters). There are 6×5 = 30 equally likely ordered outcomes.

Define
- A = "the first marble is red."
- B = "the second marble is blue."

Then A ∩ B = "first is red AND second is blue" which is exactly one ordered outcome (red, blue), so
P(A ∩ B) = 1/30.

Compute marginal probabilities:
P(A) = number of outcomes with first = red = 5/30 = 1/6.
P(B) = number of outcomes with second = blue = 5/30 = 1/6.

Check independence: P(A)P(B) = (1/6)(1/6) = 1/36 ≠ 1/30 = P(A ∩ B), so A and B are dependent (the result of the first draw affects the probabilities for the second draw).