Question

a hat contains 4 marbles; one blue, one green, one orange, and one red. two marbles are selected at random without replacement from the hat. complete parts a) through e). use the fundamental counting principle to determine the answer to part a). assume that each event is equally likely to occur.
c) determine the probability of selecting exactly 1 blue marble.
the probability is \\( \frac { 1 } { 2 } \\).
(simplify your answer.)
d) determine the probability of selecting at least 1 marble that is not blue.
the probability is 1.
(simplify your answer.)
e) determine the probability of selecting no orange marbles.
the probability is \\( \square \\)
(simplify your answer.)

Explanation:

Step1: Calculate total number of ways to select 2 marbles

Using combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 4\) (total marbles) and \(r=2\) (marbles to be selected).
\(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=\frac{4\times3}{2\times 1}=6\)

Step2: Calculate number of ways to select no orange marbles

We have 3 non - orange marbles (blue, green, red). Using combination formula with \(n = 3\) and \(r = 2\)
\(C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3\times2!}{2!\times1!}=3\)

Step3: Calculate probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
\(P=\frac{3}{6}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)