The mean of the probability distribution X is:

X

1

2

3

P(X)

0.68

0.14

0.89

a 1.285

b 3.63

C 7.71

d Cannot be determined.

To find the mean of the probability distribution \(X\), we use the formula for the expected value \(E(X)\), which is given by:

\[ E(X) = \sum (X_i \cdot P(X_i)) \]

Given:
\[ X = \{1, 2, 3\} \]
\[ P(X) = \{0.68, 0.14, 0.89\} \]

However, there is an issue here: the probabilities should sum up to 1 in a valid probability distribution. Let's check the sum of the given probabilities:

\[ 0.68 + 0.14 + 0.89 = 1.71 \]

Since the sum of the probabilities is greater than 1, this is not a valid probability distribution. Therefore, the mean cannot be determined from this information.

So, the correct answer is:

\[ d) \text{Cannot be determined.} \]

Would you like more details or have any questions? Here are some questions you might want to ask:

1. How do you find the mean of a valid probability distribution?
2. What are the properties of a probability distribution?
3. How do you handle invalid probability distributions?
4. Can you explain expected value in more detail?
5. What are some examples of valid probability distributions?

**Tip:** When working with probability distributions, always ensure that the sum of the probabilities equals 1 to confirm it is valid.

Explore how to determine the mean of a probability distribution with detailed steps and explanations. This problem examines the calculation of expected value using given probabilities and identifies issues with invalid probability distributions. Suitable for advanced high school students and learners interested in probability theory.

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