Let X denote the number of hours a person watches television during a randomly selected day. The probability that X can take the value \( x_i \), has the following form, where \( k \) is some unknown constant.
\( P(X=x_i)=\begin{cases}0.5 & \text{if }x_i=0\\kx_i & \text{if }x_i=1\\k(4-x_i) & \text{if }x_i=2\text{ or }3\\0 & \text{otherwise}\end{cases} \)
The probability that the person watches at most 2 hours of television on a selected day is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{7}{8} \)
The four probabilities are:
\( \displaystyle P(X=0)=0.5 \)
\( \displaystyle P(X=1)=k(1)=k \)
\( \displaystyle P(X=2)=k(4-2)=2k \)
\( \displaystyle P(X=3)=k(4-3)=k \)
The probabilities add up to 1:
\( \displaystyle 0.5+k+2k+k=1 \)
\( \displaystyle 4k=\frac{1}{2} \)
\( \displaystyle k=\frac{1}{8} \)
“At most 2 hours” means X ≤ 2:
\( \displaystyle P(X\le 2)=P(0)+P(1)+P(2) \)
\( \displaystyle =\frac{1}{2}+\frac{1}{8}+\frac{2}{8}=\frac{7}{8} \)
Shortcut: Use the complement: \( P(X\le 2)=1-P(X=3)=1-k=\frac{7}{8} \).
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{2} \) — Incorrect. 1/2 is only P(X = 0).
Option (3) \( \displaystyle\frac{5}{8} \) — Incorrect. 5/8 is P(X < 2) = P(0) + P(1); “at most 2” also includes X = 2.
Option (4) \( \displaystyle\frac{1}{8} \) — Incorrect. 1/8 is the value of k, which is P(X = 1) and also P(X = 3).