1. Write the given distribution

The random variable $X$ takes values $0,1,2,3,4$ with probabilities

$P(X=0)=k,\quad P(X=1)=2k,\quad P(X=2)=4k,\quad P(X=3)=6k,\quad P(X=4)=8k.$

The question text appears truncated as "The value of $P(1< X \le 4)$" (this matches the options and stored answer).

2. Use total probability = 1 to find $k$

Since probabilities sum to $1$,

$k+2k+4k+6k+8k=1$ $21k=1$ $k=\frac{1}{21}.$

3. Compute $P(1<X\le 4)$

The event $1<X\le 4$ means

$X=2,3,4.$

So,

$P(1<X\le 4)=P(X=2)+P(X=3)+P(X=4)$ $=4k+6k+8k=18k.$

Substitute $k=\frac{1}{21}$:

$18k=18\cdot \frac{1}{21}=\frac{18}{21}=\frac{6}{7}.$

4. Compare with options

None of the given options equals $\frac{6}{7}$.

However, option A is $\frac{4}{7}$, which would correspond to

$P(1\le X<4)=P(X=1)+P(X=2)+P(X=3)=2k+4k+6k=12k=\frac{12}{21}=\frac{4}{7}.$

So the printed question is likely truncated or misformatted, and the intended event was probably $P(1\le X<4)$.

Therefore:

• If the question truly asks $P(1<X\le 4)$, the answer is $\frac{6}{7}$.
• If the intended question was $P(1\le X<4)$, then the answer is $\frac{4}{7}$, matching the stored answer.