The probability distribution of a random variable X is given by
| X | 0 | 1 | 2 |
| P(X) | k | 2k | 3k |
If \(k \gt 0\), then \(P(0 \lt X \le 2)\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{5}{6}\)
The probabilities of a distribution add up to 1:
\(k + 2k + 3k = 1\)
\( \displaystyle 6k = 1,\quad k = \frac{1}{6} \)
The event \(0 \lt X \le 2\) contains X = 1 and X = 2 only:
\(P(0 \lt X \le 2) = P(X = 1) + P(X = 2)\)
\( \displaystyle = 2k + 3k = 5k = \frac{5}{6} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{6}\) — Incorrect. This is \(P(X = 0)\), the probability of the complement of the event.
Option (3) \(\displaystyle\frac{7}{8}\) — Incorrect. Every probability here is a multiple of \(\frac{1}{6}\), and \(\frac{7}{8}\) is not.
Option (4) \(\displaystyle\frac{9}{16}\) — Incorrect. Every probability here is a multiple of \(\frac{1}{6}\), and \(\frac{9}{16}\) is not.