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A bag contains 5 red marbles, 3 blue marbles and 6 green marbles. If two marbles are drawn out of the bag, what is the exact probability that both marbles drawn will be green?
Answer:

A bag contains $5$ red marbles, $3$ blue marbles and $6$ green marbles. If two marbles are drawn out of the bag, what is the exact probability that both marbles drawn will be green? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. A bag contains

5

red marbles,

3

blue marbles and

6

green marbles. If two marbles are drawn out of the bag, what is the exact probability that both marbles drawn will be green?

\newline

Answer:

Determine total number: Determine the total number of marbles in the bag. $\newline$ The bag contains $5$ red marbles, $3$ blue marbles, and $6$ green marbles. To find the total, we add these numbers together. $\newline$ $5$ (red) + $3$ (blue) + $6$ (green) = $14$ (total marbles)
Calculate first green probability: Calculate the probability of drawing the first green marble. $\newline$ The probability of an event is the number of favorable outcomes divided by the total number of outcomes. Since there are $6$ green marbles and $14$ total marbles, the probability of drawing a green marble first is: $\newline$ Probability of first green marble = Number of green marbles / Total number of marbles $\newline$ Probability of first green marble = $\frac{6}{14}$
Calculate second green probability: Calculate the probability of drawing a second green marble after the first one has been drawn. $\newline$ After drawing one green marble, there are now $5$ green marbles left and $13$ marbles in total. The probability of drawing a second green marble is: $\newline$ Probability of second green marble = Number of remaining green marbles / Total remaining marbles $\newline$ Probability of second green marble = $\frac{5}{13}$
Calculate combined probability: Calculate the combined probability of both events happening consecutively. $\newline$ To find the probability of both events happening one after the other, we multiply the probabilities of each event. $\newline$ Combined probability = Probability of first green marble $\times$ Probability of second green marble $\newline$ Combined probability = $(\frac{6}{14}) \times (\frac{5}{13})$
Simplify combined probability: Simplify the combined probability. $\newline$ We can simplify the fraction by multiplying the numerators together and the denominators together. $\newline$ Combined probability = $(6 \times 5) / (14 \times 13)$ $\newline$ Combined probability = $30 / 182$ $\newline$ This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . $\newline$ Combined probability = $30/2 / 182/2$ $\newline$ Combined probability = $15 / 91$

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