Suzy has a bag containing 3 red marbles, 2 green ones, 4 white ones, and 6 purple ones. She grabs 5 of them. Find the probability that she has 2 red ones and 1 of each of the other colors. Express the answer as a fraction in lowest terms.
2/13 for the red marbles and 3/13 for the rest of the marbles
C(3,2)*C(2,1)*C(4,1)*C(6,1)/C(15,5)
= 48/1001
To find the probability that Suzy has 2 red marbles and 1 of each of the other colors, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
First, let's find the total number of possible outcomes. Since Suzy is grabbing 5 marbles out of a bag with a total of 15 marbles, we can use combinations to calculate this. The number of combinations of choosing 5 marbles out of 15 can be represented as "15 choose 5" and can be calculated as:
C(15, 5) = 15! / (5! * (15-5)!) = 3003
Now let's find the number of favorable outcomes, which corresponds to the number of ways she can select 2 red marbles and 1 of each of the other colors.
The number of ways to select 2 red marbles out of 3 is given by "3 choose 2", which is equal to:
C(3, 2) = 3! / (2! * (3-2)!) = 3
Similarly, the number of ways to select 1 green marble out of 2, 1 white marble out of 4, and 1 purple marble out of 6 can be calculated as:
C(2, 1) = 2
C(4, 1) = 4
C(6, 1) = 6
Now, let's calculate the number of favorable outcomes by multiplying these values together:
Number of favorable outcomes = (C(3, 2)) * (C(2, 1)) * (C(4, 1)) * (C(6, 1))
= 3 * 2 * 4 * 6
= 144
Finally, we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 144 / 3003
However, the probability can be simplified by finding the greatest common divisor (GCD) of 144 and 3003, and then dividing both the numerator and denominator by the GCD.
In this case, the GCD of 144 and 3003 is 3.
So, the simplified probability is:
Probability = (144 / 3) / (3003 / 3)
= 48 / 1001
Therefore, the probability that Suzy has 2 red marbles and 1 of each of the other colors is 48/1001, expressed as a fraction in lowest terms.