The table below shows the probability distribution for a discrete random variable $X$ - AQA - A-Level Maths Pure - Question 11 - 2018 - Paper 3

To find the value of $k$, we need to ensure that the sum of all probabilities equals 1, as per the rules of probability.

The sum is given by:

$P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) = k + 2k + 4k + 2k + k$

This simplifies to:

$1k + 2k + 4k + 2k + 1k = 10k$

Setting the equation to satisfy the total probability:

$10k = 1$

Solving for $k$ gives:

$k = \frac{1}{10}$

Thus, the value of $k$ is ( \frac{1}{10} ).