A bag contains $5$ red marbles, $6$ blue marbles and $7$ green marbles. If two marbles are drawn out of the bag, what is the probability, to the nearest $10$ ooth, that both marbles drawn will be green? $\newline$ Answer:

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Q. A bag contains

5

red marbles,

6

blue marbles and

7

green marbles. If two marbles are drawn out of the bag, what is the probability, to the nearest

10

ooth, that both marbles drawn will be green?

\newline

Answer:

Calculate Total Marbles: Determine the total number of marbles in the bag. $\newline$ The bag contains $5$ red marbles, $6$ blue marbles, and $7$ green marbles. $\newline$ Total number of marbles = $5 + 6 + 7 = 18$ marbles.
Calculate Probability of First Green Marble: Calculate the probability of drawing one green marble. $\newline$ Since there are $7$ green marbles out of $18$ total marbles, the probability of drawing one green marble is: $\newline$ $P(\text{first green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{7}{18}.$
Calculate Probability of Second Green Marble: Calculate the probability of drawing a second green marble after one has already been drawn. $\newline$ After drawing one green marble, there are now $6$ green marbles left and $17$ marbles in total. $\newline$ $P(\text{second green} | \text{first green}) = \frac{\text{Number of green marbles left}}{\text{Total number of marbles left}} = \frac{6}{17}.$
Calculate Combined Probability: Calculate the combined probability of both events happening (drawing two green marbles in a row). $\newline$ The combined probability is the product of the probabilities of each individual event. $\newline$ $P(\text{both green}) = P(\text{first green}) \times P(\text{second green | first green}) = \frac{7}{18} \times \frac{6}{17}$ .
Perform Multiplication: Perform the multiplication to find the probability. $P(\text{both green}) = \frac{7}{18} \times \frac{6}{17} = \frac{42}{306}$ .
Simplify Fraction: Simplify the fraction to its lowest terms. $\newline$ $\frac{42}{306}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ . $\newline$ $P(\text{both green}) = \frac{(42 \div 6)}{(306 \div 6)} = \frac{7}{51}$ .
Convert to Decimal: Convert the probability to a decimal to find the nearest thousandth. $\newline$ $P(\text{both green}) = \frac{7}{51} \approx 0.137$ (rounded to the nearest thousandth).