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A bag contains 4 red counters and 6 green counters. Four counters are drawn at random from the bag, without replacement. State with a reason whether or not the events "at least two green counters are drawn" and "at least one counter of each colour is drawn" are independent.
It depends on whether your statement above means. If we pull a NEW set of marbles for each event, or we just pull 1 set of marbles and find the probabilities of each. If we pull just one set of 4 marbles then each event, and of course any number of events, are independent of each other.
If we pull 4 marbles and check Event A. Then we pull 4 marbles without replacement and check Event B, they are definitely not independent. The marbles we pulled out during Event A will affect the probabilities we calculate for Event B.
A: at least two green
B: at least one of each color
To show independence, you need to prove that P(A intersection B) = P(A) x P(B).
These values are easier to calculate if you remember that P(X) = 1 - P(~X).
P(B) = 1 - P(0 green) - P(4 green)
P(A) = 1 - P(0 green) - P(1 green)
P(A intersection B) = 1 - P(0 green) - P(1 green) - P(4 green).