A random variable \( x \) has the following probability distribution:
| \( X=x_i \) | 1 | 2 | 3 | 4 |
| \( P(X=x_i) \) | k | 2k | 3k | 4k |
The value of \( P(X\ge 3) \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{7}{10} \)
The probabilities of a distribution add up to 1:
\( \displaystyle k+2k+3k+4k=1 \)
\( \displaystyle 10k=1\ \Rightarrow\ k=\frac{1}{10} \)
\( \displaystyle P(X\ge 3)=P(X=3)+P(X=4) \)
\( \displaystyle =3k+4k=\frac{7}{10} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{3} \) — Incorrect. The value is \( \frac{7}{10} \), not \( \frac{1}{3} \).
Option (3) \( \displaystyle\frac{3}{10} \) — Incorrect. \( \frac{3}{10}=k+2k=P(X\le 2) \), the probability of the complementary event.
Option (4) \( \displaystyle\frac{3}{5} \) — Incorrect. The value is \( \frac{7}{10} \), not \( \frac{3}{5} \).