To determine the probability that the sequence of colors is guessed correctly on the first try, let's break down the problem step by step.
### Step 1: Identify the Total Number of Colors
The sequence must be formed using the following colors:
- Red
- Yellow
- Blue
- Purple
So, there are 4 different colors to choose from.
### Step 2: Determine the Number of Positions in the Sequence
The sequence is made up of 3 positions.
### Step 3: Calculate the Total Number of Possible Sequences
Since the order of the colors matters (i.e., permutations), we will calculate the number of different sequences possible. The formula for permutations when choosing [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] distinct items is given by:
[tex]\[ P(n, k) = \frac{n!}{(n - k)!} \][/tex]
Here, [tex]\( n = 4 \)[/tex] (total colors) and [tex]\( k = 3 \)[/tex] (positions in the sequence).
Using the factorial function [tex]\( ! \)[/tex] to compute this:
[tex]\[ P(4, 3) = \frac{4!}{(4 - 3)!} = \frac{4!}{1!} = \frac{4 \times 3 \times 2 \times 1}{1} = 24 \][/tex]
So, there are 24 unique sequences of colors possible.
### Step 4: Calculate the Probability
The probability of guessing the correct sequence on the first try is the reciprocal of the total number of possible sequences. This is because there is only one correct sequence out of the 24 possible sequences.
[tex]\[ \text{Probability} = \frac{1}{\text{Total number of possible sequences}} = \frac{1}{24} \][/tex]
Therefore, the probability that the child guesses the correct sequence on the first try is:
[tex]\[ \boxed{\frac{1}{24}} \][/tex]
