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Let's solve the problem step by step.

We have the following ten words: ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN. We are tasked with calculating:

### (a) The probability distribution for \( X \), where \( X \) is the number of letters in each word.

1. Count the number of letters for each word:
   - ONE: 3 letters
   - TWO: 3 letters
   - THREE: 5 letters
   - FOUR: 4 letters
   - FIVE: 4 letters
   - SIX: 3 letters
   - SEVEN: 5 letters
   - EIGHT: 5 letters
   - NINE: 4 letters
   - TEN: 3 letters

So, the possible values of \( X \) are 3, 4, and 5.

2. Now, calculate the probabilities for each \( X \):
   - Number of words with 3 letters: ONE, TWO, SIX, TEN (4 words)
   - Number of words with 4 letters: FOUR, FIVE, NINE (3 words)
   - Number of words with 5 letters: THREE, SEVEN, EIGHT (3 words)

   The total number of words is 10, so the probability distribution for \( X \) is:

   - \( P(X = 3) = \frac{4}{10} = 0.4 \)
   - \( P(X = 4) = \frac{3}{10} = 0.3 \)
   - \( P(X = 5) = \frac{3}{10} = 0.3 \)

### (b) The expected value \( E(X) \).

The expected value is calculated as:
\[
E(X) = \sum X \cdot P(X)
\]
Substitute the values we found:

\[
E(X) = 3 \cdot 0.4 + 4 \cdot 0.3 + 5 \cdot 0.3
\]
Now, let's calculate that.

The expected value \( E(X) \) is approximately 3.9.

### Summary:
- (a) The probability distribution for \( X \) is:
  - \( P(X = 3) = 0.4 \)
  - \( P(X = 4) = 0.3 \)
  - \( P(X = 5) = 0.3 \)
  
- (b) The expected value \( E(X) \) is 3.9.

Would you like more details or have any questions?

Here are 5 follow-up questions to expand on this:

1. How does the variance of \( X \) differ from the expected value?
2. What is the cumulative distribution function for \( X \)?
3. How would the results change if there were 12 words instead of 10?
4. What does the standard deviation of \( X \) tell us about the distribution?
5. How does altering the number of letters in one word impact the expected value?

**Tip**: The expected value gives the long-term average if we repeatedly sample the number of letters in the words.

Each of the following 10 words is placed on a card and put in a hat: ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN. We pick a card at random. Let X be the size (number of letters) of the corresponding word. (a) Give the probability distribution for X. (b) Find the expected number of X.

Explore the probability distribution and expected value for the number of letters in ten different words when selected randomly. This solution covers the basics of probability calculations and expected value computation, suitable for students in grades 7 through 12.

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