Question

Can someone please help me?

Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, three white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms.
She has all the red ones.

Answer 1

2/7 or two sevenths because there are 14 marbles total and she grabs 4, the original chances of them all being red are 4/14 but we can simplify by dividing by 2

Answer 2

Let's approach this question using a permutation or combination strategy.
Now, since we are interested in picking all four red marbles, we need to devise a mathematical equation in order to understand what it is meant by this.

By choosing all four as red, we are saying that there are only ten marbles to choose the extra four marbles, because there is only one way in choosing all four marbles.

To further demonstrate this understanding, we can approach it with a visual representation, like so:

_   _   _   _   _   _   _   _

R₁   R₂   R₃   R₄   G₁   G₂   G₃   G₄   W₁   W₂   W₃   P₁   P₂   P₃

Since we want to pick all four red marbles, we can only take the four marbles in one way. Because we have already picked four marbles as red, we only have four slots to pick the remaining ten marbles:

R₁   R₂   R₃   R₄   _   _   _   _

Thus, we can fill the remaining slots in C(10, 4) ways.

Now, this is the total number of ways in which Suzan has picked all four red marbles, and the remaining four marbles randomly. To find the probability, we need to divide our favourable conditions by the total number of ways, which can be represented as C(14, 8), or $^{14}C_8$.

Thus, our final probability is:
$\frac{^{10}C_4}{^{14}C_8} = \frac{10}{143}$