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A bag contains 5 red marbles, 8 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 $5$ red marbles, $8$ 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

5

red marbles,

8

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:

Calculate Total Marbles: Determine the total number of marbles in the bag. The bag contains $5$ red marbles, $8$ blue marbles, and $7$ green marbles. To find the total, we add these numbers together. $5$ red $+$ $8$ blue $+$ $7$ green $=$ $20$ total marbles.
First Blue Marble Probability: Calculate the probability of drawing the first blue marble. $\newline$ The probability of drawing a blue marble on the first draw is the number of blue marbles divided by the total number of marbles. $\newline$ Probability of first blue marble = Number of blue marbles / Total number of marbles = $\frac{8}{20}$ .
New Total Marbles: Determine the new total number of marbles after one blue marble has been drawn. $\newline$ After drawing one blue marble, there is one less blue marble and one less marble in total. $\newline$ New total number of marbles = $20 - 1 = 19$ marbles.
Second Blue Marble Probability: Calculate the probability of drawing a second blue marble after the first has been drawn. $\newline$ Now, there are $7$ blue marbles left and $19$ marbles in total. $\newline$ Probability of second blue marble = Number of blue marbles left / New total number of marbles = $\frac{7}{19}$ .
Consecutive Events Probability: Calculate the probability of both events happening consecutively. $\newline$ To find the probability of both marbles being blue, we multiply the probability of the first event by the probability of the second event. $\newline$ Probability of both blue marbles = Probability of first blue marble $\times$ Probability of second blue marble = $(8/20) \times (7/19)$ .
Simplify Probability: Simplify the probability. $\newline$ $(\frac{8}{20}) \times (\frac{7}{19}) = (\frac{2}{5}) \times (\frac{7}{19}) = \frac{14}{95}$ .

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