Consider the following assignments of probabilities for outcomes of sample space S = {1, 2, 3, 4, 5, 6, 7, 8}.
`{:("Number (X)",1,2,3,4,5,6,7,8),("Probability, P(X)",0.15,0.23,0.12,0.10,0.20,0.08,0.07,0.05):}`
Find the probability that
X is a prime number
(b) X is a number greater than 4.

Explanation

To solve the problem, we need to find the probabilities for two scenarios based on the given sample space \( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \) and their corresponding probabilities.

### Given Data:
- Sample Space \( S \): \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)
- Probabilities \( P(X) \):
- \( P(1) = 0.15 \)
- \( P(2) = 0.23 \)
- \( P(3) = 0.12 \)
- \( P(4) = 0.10 \)
- \( P(5) = 0.20 \)
- \( P(6) = 0.08 \)
- \( P(7) = 0.07 \)
- \( P(8) = 0.05 \)

### Part (a): Find the probability that \( X \) is a prime number.

1. **Identify the prime numbers in the sample space**:
- The prime numbers from the set \( \{1, 2, 3, 4, 5, 6, 7, 8\} \) are \( 2, 3, 5, \) and \( 7 \).

2. **Sum the probabilities of these prime numbers**:
- \( P(X \text{ is prime}) = P(2) + P(3) + P(5) + P(7) \)
- Substitute the values:
\[ P(X \text{ is prime}) = 0.23 + 0.12 + 0.20 + 0.07 \]

3. **Calculate the total**:
- \( P(X \text{ is prime}) = 0.23 + 0.12 + 0.20 + 0.07 = 0.62 \)

### Part (b): Find the probability that \( X \) is a number greater than 4.

1. **Identify the numbers greater than 4 in the sample space**:
- The numbers greater than 4 are \( 5, 6, 7, \) and \( 8 \).

2. **Sum the probabilities of these numbers**:
- \( P(X > 4) = P(5) + P(6) + P(7) + P(8) \)
- Substitute the values:
\[ P(X > 4) = 0.20 + 0.08 + 0.07 + 0.05 \]

3. **Calculate the total**:
- \( P(X > 4) = 0.20 + 0.08 + 0.07 + 0.05 = 0.40 \)

### Final Answers:
- (a) The probability that \( X \) is a prime number is \( 0.62 \).
- (b) The probability that \( X \) is a number greater than 4 is \( 0.40 \).