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A bag contains 6 red marbles, 7 blue marbles and 8 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be blue?
Answer:

A bag contains $6$ red marbles, $7$ blue marbles and $8$ green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be blue? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. A bag contains

6

red marbles,

7

blue marbles and

8

green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be blue?

\newline

Answer:

Determine total number of marbles: Determine the total number of marbles in the bag. $\newline$ The bag contains $6$ red marbles, $7$ blue marbles, and $8$ green marbles. So, the total number of marbles is $6 + 7 + 8$ .
Calculate total number of marbles: Calculate the total number of marbles. $\newline$ $6$ red + $7$ blue + $8$ green = $21$ total marbles.
Determine probability of drawing one blue marble: Determine the probability of drawing one blue marble. $\newline$ The probability of drawing one blue marble is the number of blue marbles divided by the total number of marbles, which is $\frac{7}{21}$ .
Determine probability of drawing a second blue marble: Determine the probability of drawing a second blue marble after one has already been drawn. $\newline$ After drawing one blue marble, there are now $6$ blue marbles left and $20$ total marbles. The probability of drawing a second blue marble is now $\frac{6}{20}$ .
Determine probability of drawing a third blue marble: Determine the probability of drawing a third blue marble after two have already been drawn. $\newline$ After drawing two blue marbles, there are now $5$ blue marbles left and $19$ total marbles. The probability of drawing a third blue marble is now $\frac{5}{19}$ .
Calculate probability of drawing three blue marbles in a row: Calculate the probability of drawing three blue marbles in a row. $\newline$ The probability of drawing three blue marbles consecutively is the product of the probabilities of each draw: $\newline$ $(\frac{7}{21}) \times (\frac{6}{20}) \times (\frac{5}{19})$ .
Simplify the probability: Simplify the probability. $\newline$ $(\frac{7}{21}) \times (\frac{6}{20}) \times (\frac{5}{19}) = (\frac{1}{3}) \times (\frac{3}{10}) \times (\frac{5}{19}) = (\frac{1\times3\times5}{3\times10\times19}) = \frac{5}{190} = \frac{1}{38}$ .

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