An urn contains 10 white marbles, 15 blue marbles, and 20 red marbles. Five marbles are selected, one at a time, with replacement. Find the probability that at least one color will be missing from the 5 selected marbles.
Answers
Given : An urn contains 10 white marbles, 15 blue marbles, and 20 red marbles.
Five marbles are selected, one at a time, with replacement.
To Find : the probability that at least one color will be missing from the 10 selected marbles.
Solution:
White W = 10
Blue B = 15
Red R = 20
Total = 45
Five marbles are selected, one at a time, with replacement.
P (W) - 10/45 = 2/9 => P ( not white ) = 7/9
P (B) = 15/45 = 1/3 => P ( not Blue ) = 2/3
P (R) = 20/45 = 4/9 => P ( not Red ) = 5/9
probability that at least one color will be missing from the 10 selected marbles.
At least white missing = P( no white ) = ( 7/9)⁵
At least Blue missing = P ( no blue ) = ( 2/3)⁵
At least Red missing = P ( no red) = ( 5/9)⁵
At least white missing consist of white and blue missing , white and red missing . similarly
At least Blue missing consist of white and blue missing , blue and red missing
At least Red missing consist of white and Red missing , blue and red missing
Missing of white and blue , white and red , blue and red is counted twice
P ( Missing of white and blue) = only red = (4/9)⁵
P ( Missing of white and red) = only blue = (1/3)⁵
P ( Missing of red and blue) = only white = (2/9)⁵
probability that at least one color will be missing from the 10 selected marbles.
= ( 7/9)⁵ + ( 2/3)⁵ + (5/9)⁵ - (4/9)⁵ - (1/3)⁵- (2/9)⁵
= ( 7/9)⁵ + ( 6/9)⁵ + (5/9)⁵ - (4/9)⁵- (3/9)⁵ - (2/9)⁵
= ( 7⁵ + 6⁵ + 5⁵ - 4⁵ - 3⁵- 2⁵)/9⁵
=0.44724
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Answer:
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