An urn contains 10 red marbles, 10 white marbles, and 10 blue marbles marbles. A child randomly selects three (without replacement) from the urn. Round to four decimal places. Find the probability all three marbles are the same color. Find the probability that none of the three marbles are white.
Question
An urn contains 10 red marbles, 10 white marbles, and 10 blue marbles marbles. A child randomly selects three (without replacement) from the urn. Round to four decimal places.
Find the probability all three marbles are the same color.
Find the probability that none of the three marbles are white.
Solution
Sure, let's solve these problems step by step.
- Find the probability all three marbles are the same color.
First, we need to find the total number of ways to draw 3 marbles from 30. This can be calculated using the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items, and r is the number of items to choose. So, the total number of ways to draw 3 marbles from 30 is C(30, 3) = 30! / [3!(30-3)!] = 4060.
Next, we need to find the number of ways to draw 3 marbles of the same color. Since there are 10 marbles of each color, the number of ways to draw 3 marbles of the same color is C(10, 3) = 10! / [3!(10-3)!] = 120. Since there are 3 colors, the total number of ways to draw 3 marbles of the same color is 3 * 120 = 360.
So, the probability that all three marbles are the same color is 360 / 4060 = 0.0887.
- Find the probability that none of the three marbles are white.
To find this probability, we need to consider only the red and blue marbles. The total number of ways to draw 3 marbles from the 20 red and blue marbles is C(20, 3) = 20! / [3!(20-3)!] = 1140.
So, the probability that none of the three marbles are white is 1140 / 4060 = 0.2808.
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