The discrete random variable  R*R*  is such that  R∈{2,4,5,7}*R*∈{2,4,5,7}.Given that  P(R=r)=k(r+1)r+2*P*(*R*=*r*)=*r*+2*k*(*r*+1)​, then find.

**(a)**

The value of the constant  k*k*.

**(b)**

P(R⩽4)*P*(*R*⩽4).

Note ⁣:**Note**:Give your answer in simplest fraction form only.

To solve the problem, we need to find the value of the constant \( k \) and then compute \( P(R \leq 4) \).

### (a) Finding the constant \( k \)

Given the probability mass function:
\[ P(R = r) = \frac{k(r + 1)}{r + 2} \]
where \( R \in \{2, 4, 5, 7\} \), we know that the sum of all probabilities must equal 1:
\[ \sum_{r=2,4,5,7} P(R = r) = 1 \]

Let's compute the individual probabilities:
\[ P(R = 2) = \frac{k(2 + 1)}{2 + 2} = \frac{3k}{4} \]
\[ P(R = 4) = \frac{k(4 + 1)}{4 + 2} = \frac{5k}{6} \]
\[ P(R = 5) = \frac{k(5 + 1)}{5 + 2} = \frac{6k}{7} \]
\[ P(R = 7) = \frac{k(7 + 1)}{7 + 2} = \frac{8k}{9} \]

Now, sum these probabilities:
\[ \frac{3k}{4} + \frac{5k}{6} + \frac{6k}{7} + \frac{8k}{9} = 1 \]

To combine these fractions, find a common denominator. The least common multiple of 4, 6, 7, and 9 is 252. Convert each fraction:
\[ \frac{3k}{4} = \frac{189k}{252} \]
\[ \frac{5k}{6} = \frac{210k}{252} \]
\[ \frac{6k}{7} = \frac{216k}{252} \]
\[ \frac{8k}{9} = \frac{224k}{252} \]

Summing these:
\[ \frac{189k}{252} + \frac{210k}{252} + \frac{216k}{252} + \frac{224k}{252} = 1 \]
\[ \frac{839k}{252} = 1 \]

Solving for \( k \):
\[ k = \frac{252}{839} \]

So, the value of the constant \( k \) is:
\[ k = \frac{252}{839} \]

### (b) Finding \( P(R \leq 4) \)

\[ P(R \leq 4) = P(R = 2) + P(R = 4) \]

Using the probabilities calculated with \( k = \frac{252}{839} \):
\[ P(R = 2) = \frac{3k}{4} = \frac{3 \times \frac{252}{839}}{4} = \frac{756}{3356} = \frac{189}{839} \]
\[ P(R = 4) = \frac{5k}{6} = \frac{5 \times \frac{252}{839}}{6} = \frac{1260}{5034} = \frac{210}{839} \]

Sum these probabilities:
\[ P(R \leq 4) = \frac{189}{839} + \frac{210}{839} \]
\[ P(R \leq 4) = \frac{399}{839} \]

So, the value of \( P(R \leq 4) \) is:
\[ P(R \leq 4) = \frac{399}{839} \]

To summarize:
(a) The value of the constant \( k \) is \( \frac{252}{839} \).
(b) \( P(R \leq 4) = \frac{399}{839} \).

Do you want details or have any questions?

Here are 5 numbered questions for you to ask:

1. How do you compute the sum of probabilities for a discrete random variable?
2. What steps are involved in finding a common denominator for adding fractions?
3. How can you simplify fractions to their simplest form?
4. Why is it important for the sum of probabilities to equal 1?
5. How do you interpret the result of a probability calculation?

**Tip:** When solving probability problems, always double-check your work to ensure that the sum of the probabilities equals 1, as this is a fundamental property of probability distributions.

Explore how to solve a probability problem involving a discrete random variable with given probabilities and constants. Learn the steps to find the constant \( k \) and compute \( P(R \leq 4) \) in simplest fraction form. Suitable for high school students learning probability theory.

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