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A bag contains 5 red marbles, 4 blue marbles and 7 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red?
Answer:

A bag contains $5$ red marbles, $4$ blue marbles and $7$ green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. A bag contains

5

red marbles,

4

blue marbles and

7

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

\newline

Answer:

Determine total number of marbles: Determine the total number of marbles in the bag. $\newline$ The bag contains $5$ red marbles, $4$ blue marbles, and $7$ green marbles. So, the total number of marbles is $5 + 4 + 7 = 16$ marbles.
Calculate probability of drawing one red marble: Calculate the probability of drawing one red marble. The probability of drawing one red marble is the number of red marbles divided by the total number of marbles, which is $\frac{5}{16}$ .
Calculate probability of drawing second red marble: Calculate the probability of drawing a second red marble after one has already been drawn. $\newline$ After drawing one red marble, there are now $4$ red marbles left and $15$ marbles in total. The probability of drawing a second red marble is now $\frac{4}{15}$ .
Calculate probability of drawing third red marble: Calculate the probability of drawing a third red marble after two have already been drawn. $\newline$ After drawing two red marbles, there are now $3$ red marbles left and $14$ marbles in total. The probability of drawing a third red marble is now $\frac{3}{14}$ .
Calculate probability of all three events: Calculate the probability of all three events happening in sequence (drawing three red marbles in a row). $\newline$ The probability of all three events happening in sequence is the product of the probabilities of each event. This is calculated as $(\frac{5}{16}) \times (\frac{4}{15}) \times (\frac{3}{14})$ .
Perform multiplication to find exact probability: Perform the multiplication to find the exact probability. $\newline$ The exact probability is $(\frac{5}{16}) \times (\frac{4}{15}) \times (\frac{3}{14}) = (\frac{5 \times 4 \times 3}{16 \times 15 \times 14}) = \frac{60}{3360}$ .
Simplify fraction to lowest terms: Simplify the fraction to its lowest terms. $\newline$ The fraction $\frac{60}{3360}$ can be simplified by dividing both the numerator and the denominator by the greatest common divisor, which is $60$ . So, $\frac{60}{3360}$ simplifies to $\frac{1}{56}$ .

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