Question: A jar contains 3 white marbles, 2 yellow marbles, 4 red marbles, and 5 blue marbles. Two marbles are picked at random. What is the probability that

a.) Both are blue
b.) Exactly 1 is blue
c.) At least 1 is blue

Solution This solution is FREE courtesy of BrainMass!

(a) The event of choosing 2 marbles has:
14!/(12!2!) = 91 members, corresponding to picking 2 objects out of 14.
The event of choosing 2 blue ones corresponds to choosing 2 blue marbles among 5: 5!/2!3! = 10.
The probability is given by the ratio of these two: 10/91.

(b) This time our event corresponds to picking 1 blue out of 5 (5 choices) and then picking one of the other colors (14-5=9 choices). Overall we have 5x9 = 45 choices. The probability is given by: 45/91.

(c) The easiest way to get this is to observe that this probability can be deduced as the sum of probability of getting 2 and getting 1 exactly, which are just what we calculated in (a) and (b). Therefore the probability is: 10/91 + 45/91= 55/91.

Note: That we couldn't have chosen 1 blue ball first (5) and then pick another from the remaining 13 (65). This way we are double counting the case where the first one is not blue but the second one is (10).

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