Of course! Let's solve the problem step-by-step according to the provided hint.
Step 1: Determine the number of ways to choose 3 Jolly Ranchers out of 15.
This can be calculated using the combination formula, which determines how many ways you can choose [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items without regard to order. The combination formula is [tex]\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)[/tex].
For our problem, [tex]\( n = 15 \)[/tex] and [tex]\( k = 3 \)[/tex]:
[tex]\[
n(S) = \binom{15}{3}
\][/tex]
According to the given information, we already know:
[tex]\[
n(S) = 455
\][/tex]
Step 2: Determine the number of favorable ways to choose 3 blue Jolly Ranchers out of the 3 blue Jolly Ranchers available in the jar.
Again, we use the combination formula, but here [tex]\( n = 3 \)[/tex] and [tex]\( k = 3 \)[/tex]:
[tex]\[
n(E) = \binom{3}{3}
\][/tex]
According to the given information, we already know:
[tex]\[
n(E) = 1
\][/tex]
Step 3: Calculate the probability as the ratio of the number of favorable outcomes to the total number of outcomes.
The probability [tex]\( P \)[/tex] that all 3 chosen Jolly Ranchers are blue is given by:
[tex]\[
P = \frac{n(E)}{n(S)}
\][/tex]
Substitute the values we have:
[tex]\[
P = \frac{1}{455}
\][/tex]
Therefore, the probability that all 3 Jolly Ranchers chosen are blue is:
[tex]\[
\boxed{\frac{1}{455}}
\][/tex]
