A bag contains 2 red marbles, 8 blue marbles, and 5 green marbles. If two marbles are drawn out of the bag, what is the probability, to the nearest 10th of a percent, that both mar... A bag contains 2 red marbles, 8 blue marbles, and 5 green marbles. If two marbles are drawn out of the bag, what is the probability, to the nearest 10th of a percent, that both marbles drawn will be green?
Understand the Problem
The question is asking for the probability of drawing two green marbles from a bag containing a total of 15 marbles, which includes 2 red, 8 blue, and 5 green marbles. To solve this, we'll first find the total combinations for drawing two marbles and then the combinations for drawing two green marbles, finally calculating the probability from these figures.
Answer
The probability of drawing two green marbles is $\frac{2}{21}$.
Answer for screen readers
The probability of drawing two green marbles is $\frac{2}{21}$.
Steps to Solve
- Calculate Total Combinations of Drawing Two Marbles
To find the total ways to choose 2 marbles from 15, we use the combination formula:
$$ C(n, r) = \frac{n!}{r!(n-r)!} $$
where $n$ is the total number of marbles and $r$ is the number of marbles drawn.
Here, $n = 15$ and $r = 2$:
$$ C(15, 2) = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 $$
- Calculate Combinations of Drawing Two Green Marbles
Now, we calculate the combinations specifically for selecting 2 green marbles from the 5 available green marbles:
$$ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 $$
- Calculate the Probability of Drawing Two Green Marbles
The probability is the ratio of the combinations of drawing two green marbles to the total combinations of drawing two marbles:
$$ P(\text{two green}) = \frac{C(5, 2)}{C(15, 2)} = \frac{10}{105} $$
To simplify:
$$ P(\text{two green}) = \frac{2}{21} $$
The probability of drawing two green marbles is $\frac{2}{21}$.
More Information
In probability, combinations allow us to determine how many ways we can select items from a larger set without regard to the order of selection. This situation reflects everyday scenarios, like drawing marbles from a bag, which can be linked to topics in statistics.
Tips
- Not using the combination formula correctly. Ensure that you use $C(n, r)$ for combinations instead of permutations.
- Forgetting to simplify the final probability fraction. Always check if the result can be reduced to its simplest form.
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