A random variable $$X$$ has the probability distribution: $$X: 1, 2, 3, 4, 5, 6, 7, 8$$; $$p(X): 0.15, 0.23, 0.12, 0.10, 0.20, 0.08, 0.07, 0.05$$. For the events $$E = \{X$$ is a prime number$$\}$$ and $$F = \{X < 4\}$$, the probability $$P(E \cup F)$$ is

We are given the probability distribution for a random variable $$ X $$:

$$ X = 1 \implies P(1) = 0.15 $$

$$ X = 2 \implies P(2) = 0.23 $$

$$ X = 3 \implies P(3) = 0.12 $$

$$ X = 4 \implies P(4) = 0.10 $$

$$ X = 5 \implies P(5) = 0.20 $$

$$ X = 6 \implies P(6) = 0.08 $$

$$ X = 7 \implies P(7) = 0.07 $$

$$ X = 8 \implies P(8) = 0.05 $$

Step 1: Identify the Outcomes for Event $$ E $$

Event $$ E $$ consists of outcomes where $$ X $$ is a prime number. The prime numbers in the given set are 2, 3, 5, and 7:

$$ E = \{2, 3, 5, 7\} $$

Step 2: Identify the Outcomes for Event $$ F $$

Event $$ F $$ consists of outcomes where $$ X < 4 $$:

$$ F = \{1, 2, 3\} $$

Step 3: Find the Union of Events $$ E $$ and $$ F $$

The event $$ E \cup F $$ includes all unique outcomes belonging to either $$ E $$, $$ F $$, or both:

$$ E \cup F = \{1, 2, 3, 5, 7\} $$

Step 4: Calculate the Combined Probability

To find $$ P(E \cup F) $$, we add up the individual probabilities of each outcome in the union:

$$ P(E \cup F) = P(1) + P(2) + P(3) + P(5) + P(7) $$

$$ P(E \cup F) = 0.15 + 0.23 + 0.12 + 0.20 + 0.07 $$

$$ P(E \cup F) = 0.77 $$

Therefore, the probability $$ P(E \cup F) $$ is $$ 0.77 $$.