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A bag contains 2 red marbles, 5 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 $2$ red marbles, $5$ 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

2

red marbles,

5

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:

Calculate Total Marbles: Determine the total number of marbles in the bag. The bag contains $2$ red marbles, $5$ blue marbles, and $6$ green marbles. To find the total, we add these numbers together. $2$ (red) + $5$ (blue) + $6$ (green) = $13$ (total marbles)
Probability of First Green: 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. For the first draw, the number of favorable outcomes is the number of green marbles, and the total number of outcomes is the total number of marbles in the bag. $\newline$ Probability of first green marble = Number of green marbles / Total number of marbles = $\frac{6}{13}$
New Total Marbles: Determine the new total number of marbles in the bag after one green marble has been drawn. $\newline$ After drawing one green marble, there is one less green marble and one less marble overall in the bag. $\newline$ New total number of marbles = $13 - 1 = 12$
Probability of Second Green: Calculate the probability of drawing the second green marble after the first has been drawn. $\newline$ Now, the number of favorable outcomes is the remaining number of green marbles, and the total number of outcomes is the new total number of marbles in the bag. $\newline$ Probability of second green marble = Remaining number of green marbles / New total number of marbles = $\frac{5}{12}$
Probability of Both Events: Calculate the probability of both events happening together (drawing two green marbles in a row). $\newline$ To find the probability of both events happening, we multiply the probability of the first event by the probability of the second event. $\newline$ Probability of both green marbles = Probability of first green marble $\times$ Probability of second green marble = $(\frac{6}{13}) \times (\frac{5}{12})$
Perform Multiplication: Perform the multiplication to find the exact probability. $(\frac{6}{13}) \times (\frac{5}{12}) = \frac{30}{156}$
Simplify Fraction: Simplify the fraction to its lowest terms. $\newline$ $\frac{30}{156}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ . $\newline$ $30 \div 6 = 5$ $\newline$ $156 \div 6 = 26$ $\newline$ So, the simplified fraction is $\frac{5}{26}$ .

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