A random variable X has the following probability distribution :
`{:(X:,1,2,3,4,5,6,7,8),(P(X):,0.15,0.23,0.12,0.10,0.20,0.08,0.07,0.08):}`
for the events `E=`[X is a prime number] `F={Xlt 4}`, the probability `P(EOP)`, is.

Options

A. `0.50`
B. `0.77`
C. `0.35`
D. `0.87`

Explanation

To solve the problem, we need to find the probability of the union of two events: E (where X is a prime number) and F (where X is less than 4). We will follow these steps:

### Step 1: Identify the Prime Numbers
First, we need to identify which values of X are prime numbers. The prime numbers from the set {1, 2, 3, 4, 5, 6, 7, 8} are:
- 2
- 3
- 5
- 7

### Step 2: Calculate the Probability of Event E
Next, we calculate the probability of event E, which is the sum of the probabilities of X being a prime number:
- P(X = 2) = 0.23
- P(X = 3) = 0.12
- P(X = 5) = 0.20
- P(X = 7) = 0.07

Now we add these probabilities:
\[ P(E) = P(X = 2) + P(X = 3) + P(X = 5) + P(X = 7) \]
\[ P(E) = 0.23 + 0.12 + 0.20 + 0.07 = 0.62 \]

### Step 3: Calculate the Probability of Event F
Now, we calculate the probability of event F, which is the sum of the probabilities of X being less than 4:
- P(X = 1) = 0.15
- P(X = 2) = 0.23
- P(X = 3) = 0.12

Now we add these probabilities:
\[ P(F) = P(X = 1) + P(X = 2) + P(X = 3) \]
\[ P(F) = 0.15 + 0.23 + 0.12 = 0.50 \]

### Step 4: Calculate the Intersection of Events E and F
Next, we need to find the intersection of events E and F, which is the probability that X is both a prime number and less than 4. The prime numbers less than 4 are:
- 2
- 3

Now we add the probabilities:
\[ P(E \cap F) = P(X = 2) + P(X = 3) \]
\[ P(E \cap F) = 0.23 + 0.12 = 0.35 \]

### Step 5: Calculate the Probability of E Union F
Finally, we use the formula for the probability of the union of two events:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
Substituting the values we found:
\[ P(E \cup F) = 0.62 + 0.50 - 0.35 \]
\[ P(E \cup F) = 0.77 \]

### Final Answer
Thus, the probability \( P(E \cup F) \) is **0.77**.

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