To determine the probability that the sequence is guessed correctly on the first try, let's break down the problem step by step.
1. Identify the number of available colors:
There are 4 different colors available: red, yellow, blue, and purple.
2. Determine the length of the sequence:
The sequence to be guessed consists of 3 colors.
3. Understand the constraints:
Each color can only be used once in the sequence, meaning there are no repetitions in a given sequence.
4. Calculate the total number of possible sequences:
To compute this, we need to consider the permutations of 4 colors taken 3 at a time. The formula for permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time is given by:
[tex]\[
P(n, r) = \frac{n!}{(n-r)!}
\][/tex]
Plugging in our values [tex]\( n = 4 \)[/tex] and [tex]\( r = 3 \)[/tex]:
[tex]\[
P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 4 \cdot 3 \cdot 2 \cdot 1 = 24
\][/tex]
Therefore, there are 24 different possible sequences of three colors.
5. Calculate the probability of guessing the correct sequence on the first try:
The probability of guessing the correct sequence is the reciprocal of the total number of possible sequences since there is only one correct sequence:
[tex]\[
\text{Probability} = \frac{1}{24}
\][/tex]
Therefore, the probability that the sequence is guessed correctly on the first try is:
[tex]\(\boxed{\frac{1}{24}}\)[/tex]
