Anindya Madhudor wrote:A box contains three pairs of blue gloves and two pairs of green gloves. Each pair consists of a left-hand glove and a right-hand glove. Each of the gloves is separate from its mate and thoroughly mixed together with the others in the box. If three gloves are randomly selected from the box, what is the probability that a matched set (i.e., a left- and right-hand glove of the same color) will be among the three gloves selected?
a. 3/10
b. 23/60
c. 7/12
d. 41/60
e. 5/6
OA: D
P(matching set) = 1 - P(no matching set).
There are 3 ways to NOT select a matching set of gloves.
Case 1: 3 non-matching blue
P(1st glove is blue) = 6/10.
P(2nd glove is blue and for the same hand) = 2/9.
P(3rd glove is blue and for the same hand) = 1/8.
Since we want all of these events to happen, we multiply the fractions:
6/10 * 2/9 * 1/8 = 1/60.
Case 2: 1 green, 2 non-matching blue
P(1st glove is green) = 4/10.
P(2nd glove is blue) = 6/9.
P(3rd glove is blue and for the same hand as the 2nd glove) = 2/8.
Since we want all of these events to happen, we multiply the fractions:
4/10 * 6/9 * 2/8 = 4/60.
Since the one green glove could be 1st, 2nd or 3rd, we multiply by 3:
4/60 * 3 = 12/60.
Case 3: 1 blue, 2 non-matching green
P(1st glove is blue) = 6/10.
P(2nd glove is green) = 4/9.
P(3rd glove is green and for the same hand as the 2nd glove) = 1/8.
Since we want all of these events to happen, we multiply the fractions:
6/10 * 4/9 * 1/8 = 1/30.
Since the one blue glove could be 1st, 2nd or 3rd, we multiply by 3:
1/30 * 3 = 3/30 = 6/60.
Since any of the 3 cases above would yield no matching set of gloves, we add the probabilities:
P(no matching set) = 1/60 + 12/60 + 6/60 = 19/60.
Thus, P(matching set) = 1 - 19/60 = 41/60.
The correct answer is
D.
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