Please check the solution below:

X, is assumed to be a random variable by placing letters of the word “YACHT” in a hat. If these letters are withdrawn then it gets replaced. A value of 1 is given when a vowel is retrieved, and a value of 2 is given when a consonant is retrieved while a value of 3 is given to all other retrieved letters.

Write down the sample space for X variable.

Can we write the sample space as given below:

S = {Y3, A1, C2, H2 and T2}.

The sample space is defined as the set of all possible outcomes of an experiment or random process. In this case, the experiment is drawing letters from a hat, and the random variable X represents the value assigned to each letter.

To find the sample space for the X variable, we need to consider all possible letters that can be drawn from the hat and the corresponding values assigned to them.

The word "YACHT" has 5 distinct letters: Y, A, C, H, and T. According to the given conditions, each time a letter is drawn, it is replaced, so the same letters can be drawn repeatedly.

Now, let's determine the assigned values for each letter:

- Y: The letter Y is a consonant, so it is assigned a value of 2. Therefore, Y corresponds to the outcome Y2.
- A: The letter A is a vowel, so it is assigned a value of 1. Hence, A corresponds to the outcome A1.
- C: C is a consonant, and it is assigned a value of 2. So, C corresponds to the outcome C2.
- H: H is a consonant, and it is also assigned a value of 2. Therefore, H corresponds to the outcome H2.
- T: T is a consonant as well, so it is assigned a value of 2. Hence, T corresponds to the outcome T2.

Therefore, the sample space for the X variable can be written as:

S = {Y2, A1, C2, H2, T2}

So, the sample space you provided: {Y3, A1, C2, H2, T2} is incorrect because the value assigned to the letter Y should be 2, not 3.

Hope this clears up any confusion regarding the correct sample space.