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A bag contains 4 red marbles, 3 blue marbles and 7 green marbles. If two marbles are drawn out of the bag, what is the exact probability that both marbles drawn will be blue?
Answer:

A bag contains $4$ red marbles, $3$ blue marbles and $7$ green marbles. If two marbles are drawn out of the bag, what is the exact probability that both marbles drawn will be blue? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. A bag contains

4

red marbles,

3

blue marbles and

7

green marbles. If two marbles are drawn out of the bag, what is the exact probability that both 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 $4$ red marbles, $3$ blue marbles, and $7$ green marbles. So, the total number of marbles is $4 + 3 + 7$ .
Calculate total number: Calculate the total number of marbles. $\newline$ $4$ red + $3$ blue + $7$ green = $14$ marbles in total.
Determine probability of first blue marble: Determine the probability of drawing the first blue marble. The probability of drawing a blue marble on the first draw is the number of blue marbles divided by the total number of marbles. Probability of first blue marble = Number of blue marbles / Total number of marbles = $\frac{3}{14}$ .
Determine new total number: Determine the new total number of marbles after one blue marble is drawn. After drawing one blue marble, there are $2$ blue marbles left and the total number of marbles is now $13$ (since one marble has been removed and not replaced).
Determine probability of second blue marble: Determine the probability of drawing the second blue marble. The probability of drawing a second blue marble after the first one has been drawn is the number of remaining blue marbles divided by the new total number of marbles. Probability of second blue marble = Remaining blue marbles / New total number of marbles = $\frac{2}{13}$ .
Calculate combined probability: Calculate the combined probability of both events happening consecutively. $\newline$ The probability of both events (drawing two blue marbles in a row) is the product of the probabilities of each individual event. $\newline$ Combined probability = Probability of first blue marble $\times$ Probability of second blue marble = $(\frac{3}{14}) \times (\frac{2}{13})$ .
Perform multiplication: Perform the multiplication to find the exact probability. $\newline$ Combined probability = $(\frac{3}{14}) \times (\frac{2}{13}) = \frac{6}{182}$ .
Simplify fraction: Simplify the fraction to its lowest terms. $\newline$ $\frac{6}{182}$ can be simplified by dividing both the numerator and the denominator by the greatest common divisor, which is $6$ . $\newline$ $\frac{6}{182} = \frac{1}{30.3}$ .

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