A bag contains 4 red marbles, 3 blue marbles, and 6 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red... A bag contains 4 red marbles, 3 blue marbles, and 6 green marbles. If three marbles are drawn out of the bag, what is the exact probability that all three marbles drawn will be red? (in decimal form)
Understand the Problem
The question is asking us to calculate the probability of drawing three red marbles from a bag containing a total of 13 marbles. We need to use the formula for probability, which is the number of favorable outcomes divided by the total number of outcomes. In this case, we need to calculate the combinations of drawing 3 red marbles from the 4 available and the combinations of drawing 3 marbles from the 13 total marbles.
Answer
The probability is $P(\text{3 red}) = \frac{2}{143}$.
Answer for screen readers
The probability of drawing 3 red marbles from the bag is $P(\text{3 red}) = \frac{2}{143}$.
Steps to Solve
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Identify Favorable Outcomes We have 4 red marbles in the bag. We need to find the number of ways to choose 3 red marbles from these 4. This can be calculated using the combination formula: $$ C(n, r) = \frac{n!}{r!(n - r)!} $$ Here, ( n = 4 ) (red marbles) and ( r = 3 ).
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Calculate Favorable Outcomes Using the formula for combinations: $$ C(4, 3) = \frac{4!}{3!(4 - 3)!} = \frac{4!}{3! \cdot 1!} = \frac{4}{1} = 4 $$ So, there are 4 ways to choose 3 red marbles.
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Identify Total Outcomes Next, we need to determine the total number of ways to choose any 3 marbles from the 13 available. Again using the combination formula: $$ C(13, 3) = \frac{13!}{3!(13 - 3)!} $$ Here, ( n = 13 ) and ( r = 3 ).
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Calculate Total Outcomes Calculating the total outcomes: $$ C(13, 3) = \frac{13!}{3! \cdot 10!} = \frac{13 \cdot 12 \cdot 11}{3 \cdot 2 \cdot 1} = \frac{1716}{6} = 286 $$ Thus, there are 286 ways to choose 3 marbles from 13.
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Calculate Probability Now, we can calculate the probability of drawing 3 red marbles: $$ P(\text{3 red}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{286} $$
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Simplify the Probability To simplify the probability: $$ P(\text{3 red}) = \frac{2}{143} $$
The probability of drawing 3 red marbles from the bag is $P(\text{3 red}) = \frac{2}{143}$.
More Information
This result shows that drawing 3 red marbles from a mixture of colors is relatively unlikely. The probability of $\frac{2}{143}$ indicates that for every 145 times you may try drawing 3 marbles, 2 of those attempts will yield all red marbles if the experiment were repeated many times.
Tips
- Forgetting to use the combination formula: Make sure to count combinations correctly using $C(n, r)$.
- Confusing combinations with permutations: Remember, combinations do not consider order, while permutations do.
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