8. An angle has the following probability distribution: 𝑓 𝜃 = (1 + sin 3𝜃 )/2𝜋.
What is the chance that the angle will lie between 0 and 120°?
a. 1/3
b. 0.5
c. 𝜋/4
d. 1
Probability Angle 0 to 120 Degrees
The probability density function for the angle θ is f(θ) = (1 + sin 3θ)/(2π), where θ ranges from 0 to 360° (or 0 to 2π radians). The probability that θ lies between 0° and 120° is the integral ∫ from 0 to 120° f(θ) dθ, which evaluates exactly to 1/3.
Solution Steps
Convert limits to radians: 0 to 2π/3. The integral becomes ∫ from 0 to 2π/3 (1 + sin 3θ)/(2π) dθ = (1/(2π)) [θ - (1/3) cos 3θ] from 0 to 2π/3.
Evaluate: (1/(2π)) [(2π/3 - (1/3)(-1/2)) - (0 - (1/3)(1))] = 1/3.
Numerical verification confirms the value 0.333333 with negligible error, and the total integral over 0° to 360° is 1, validating the PDF.
Option Analysis
- a. 1/3: Correct, as the exact integral yields precisely 1/3.
- b. 0.5: Incorrect; this would imply half the circle, but the sin(3θ) modulation skews density unevenly.
- c. π/4 ≈ 0.785: Incorrect; exceeds 1/3 and ignores the PDF’s normalization.
- d. 1: Incorrect; represents the full range (0°-360°), not 0°-120°.
Key Integral Derivation
∫ f(θ) dθ = (1/(2π)) (θ - (1/3) cos 3θ)Evaluate at
2π/3 radians minus 0 yields 1/3 precisely, due to trigonometric identity simplification.Why Not Other Options?
Uniform distribution (no sin term) gives 120/360 = 1/3 coincidentally, but sin 3θ triples frequency, concentrating probability every 120° symmetrically—still integrating to 1/3 over each third.
Option Values Table
| Option | Value | Reason |
|---|---|---|
| a. 1/3 | 0.333 | Exact match |
| b. 0.5 | 0.500 | Ignores modulation skew |
| c. π/4 | 0.785 | Unrelated constant |
| d. 1 | 1.000 | Full domain only |
Exam Tips
- Verify PDF integrates to 1 over full range.
- Use radian substitution for trig integrals.
- Triple-angle sin 3θ implies 120° periodicity—explains even thirds.
