To solve the given problem, we'll use the same approach as before. The discrete random variable $R$ takes on the values $\{1, 2, 4, 5\}$ with probabilities defined by $P(R = r) = \frac{k(r+1)}{r+2}$.

Part (a) Find the value of the constant $k$.

Since $R$ is a discrete random variable, the sum of all probabilities must equal 1:

$\sum_{r \in \{1, 2, 4, 5\}} P(R = r) = 1$

Let's compute each probability:

$P(R = 1) = \frac{k(1+1)}{1+2} = \frac{2k}{3}$

$P(R = 2) = \frac{k(2+1)}{2+2} = \frac{3k}{4}$

$P(R = 4) = \frac{k(4+1)}{4+2} = \frac{5k}{6}$

$P(R = 5) = \frac{k(5+1)}{5+2} = \frac{6k}{7}$

Now, sum these probabilities and set them equal to 1:

$\frac{2k}{3} + \frac{3k}{4} + \frac{5k}{6} + \frac{6k}{7} = 1$

To combine these fractions, find a common denominator. The least common multiple of 3, 4, 6, and 7 is 84:

$\frac{2k \cdot 28}{84} + \frac{3k \cdot 21}{84} + \frac{5k \cdot 14}{84} + \frac{6k \cdot 12}{84} = 1$

$\frac{56k}{84} + \frac{63k}{84} + \frac{70k}{84} + \frac{72k}{84} = 1$

Combine the numerators:

$\frac{56k + 63k + 70k + 72k}{84} = 1$

$\frac{261k}{84} = 1$

Solving for $k$:

$k = \frac{84}{261} = \frac{4}{13}$

Part (b) Find $P(R \leq 3)$.

Since $R$ can only be $1$ or $2$ and both are $\leq 3$:

$P(R \leq 3) = P(R = 1) + P(R = 2)$

Substitute the value of $k$:

$P(R = 1) = \frac{2k}{3} = \frac{2 \cdot \frac{4}{13}}{3} = \frac{8}{39}$

$P(R = 2) = \frac{3k}{4} = \frac{3 \cdot \frac{4}{13}}{4} = \frac{12}{52} = \frac{3}{13}$

Adding these probabilities:

$P(R \leq 3) = \frac{8}{39} + \frac{3}{13}$

To combine these fractions, find a common denominator. The least common multiple of 39 and 13 is 39:

$\frac{8}{39} + \frac{3 \cdot 3}{13 \cdot 3} = \frac{8}{39} + \frac{9}{39} = \frac{17}{39}$

So, the answers are:

• (a) $k = \frac{4}{13}$
• (b) $P(R \leq 3) = \frac{17}{39}$

Would you like further details on any part of this solution or have any questions?

Here are some additional questions you might find useful:

1. How do you find the least common multiple of denominators?
2. What is the significance of the sum of probabilities equaling 1 in a probability distribution?
3. How do you convert a probability fraction into a decimal?
4. Can you explain more about discrete random variables and their properties?
5. How can we check if a probability distribution is valid?

Tip: Always double-check your calculations, especially when finding a common denominator and combining fractions.