marbles: A bag contains 4 blue marbles, 3 white marbles, and

[jillgirl15]

A bag contains 4 blue marbles, 3 white marbles, and 6 red marbles. Three marbles are selected from the bag. What is the probability they are all red? All of the same color?

 

[soroban]

Re: marbles

Hello, jillgirl15!

A bag contains 4 blue marbles, 3 white marbles, and 6 red marbles.
Three marbles are selected from the bag (without replacement).
(a) What is the probability they are all red?
(b) All of the same color?

Three marbles are selected from the available 13 marbles.
$\displaystyle \;\;$There are: $\displaystyle \,\begin{pmatrix}13\\3\end{pmatrix}\,=\,286$ possible selections.

(a) To select 3 Red from the available 6 Red, there are: $\displaystyle \,\begin{pmatrix}6\\3\end{pmatrix}\,=\,20$ ways.
Therefore: \(\displaystyle \,P(\text{all Red) \:=\:\frac{20}{286}\:=\:\L\frac{10}{143}\)


(b) To select 3 Red from the available 6 Red, there are: $\displaystyle 20$ ways.
To select 3 White from the available 3 White, there is: $\displaystyle \begin{pmatrix}3\\3\end{pmatrix}\,=\,1$ way.
To select 3 Blue from the available 4 Blue, there are: $\displaystyle \begin{pmatrix}4\\3\end{pmatrix} \,= \,4$ ways.
$\displaystyle \;\;$Hence, there are: $\displaystyle \,20\,+\,1\,+\,4\:=\:25$ ways to get three of the same color.

Therefore: \(\displaystyle \,P(\text{same color}) \:= \:\L\frac{25}{286}\)