Boxes and marbles

Probability Level 3

There are three boxes that contains marbles. The first box has 4 yellow, 6 red, and 7 blue marbles. The second box has 8 yellow, 5 green, and 6 red marbles. The third box has 6 green, 4 red, and 8 blue marbles. A box is selected at random then 2 marbles are drawn together. Find the probability that the two marbles drawn are of the same color.

Give your answer to 3 decimal places.


The answer is 0.313.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

a. The first box is selected, then the two marbles drawn are of the same color. \text{a. The first box is selected, then the two marbles drawn are of the same color.} The probability of selecting the first box is \text{The probability of selecting the first box is} 1 3 , \frac{1}{3}, so we have \text{so we have}

P a = 1 3 ( 4 C 2 + 6 C 2 + 7 C 2 17 C 2 ) = 7 68 \large P_a = \frac{1}{3}(\frac{4C2+6C2+7C2}{17C2}) = \frac{7}{68}

b. The second box is selected, then the two marbles drawn are of the same color. \text{b. The second box is selected, then the two marbles drawn are of the same color.} The probability of selecting the second box is \text{The probability of selecting the second box is} 1 3 , \frac{1}{3}, so we have \text{so we have}

P b = 1 3 ( 8 C 2 + 6 C 2 + 5 C 2 19 C 2 ) = 53 513 \large P_b = \frac{1}{3}(\frac{8C2+6C2+5C2}{19C2}) = \frac{53}{513}

c. The third box is selected, then the two marbles drawn are of the same color. \text{c. The third box is selected, then the two marbles drawn are of the same color.} The probability of selecting the third box is \text{The probability of selecting the third box is} 1 3 , \frac{1}{3}, so we have \text{so we have}

P c = 1 3 ( 6 C 2 + 4 C 2 + 8 C 2 18 C 2 ) = 49 459 \large P_c = \frac{1}{3}(\frac{6C2+4C2+8C2}{18C2}) = \frac{49}{459}

∴ \large\therefore P ( a o r b o r c ) = 7 68 + 53 513 + 49 459 = \large P_{(a~or~b~or~c)} = \frac{7}{68} + \frac{53}{513} + \frac{49}{459} = 0.313 \boxed{\color{#69047E}\large 0.313}

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