* : In a musical chair game, the person playing the music has been advised to stop playing the music at any time within 2 minutes after she starts playing. What is the probability that the music will stop within the first half-minute after starting?
Step-by-Step Solution
Key Concept: When an event can occur at any instant uniformly over a given interval, the probability of it occurring in a sub‑interval is the ratio of the length of the sub‑interval to the length of the whole interval. This uses the concept of uniform probability distribution on a continuous interval.
1. Identify the sample space\\
The music can stop at any time \(t\) such that \(0 \le t \le 2\) minutes. Hence the sample space \(S\) is the interval \([0,2]\) minutes.
2. Assume uniform distribution\\
Since the stopping time is equally likely at any instant within the 2‑minute interval, the probability density is constant. For a uniform distribution on \([0,2]\), the total probability is 1.
3. Determine the favourable interval\\
The event of interest is "music stops within the first half‑minute" i.e. \(0 \le t \le 0.5\) minutes. The length of this favourable interval is \(0.5\) minute.
4. Compute the probability\\
\[
P(\text{stop within first 0.5 min}) = \frac{\text{length of favourable interval}}{\text{length of total interval}} = \frac{0.5}{2} = \frac{1}{4}.
\]
5. Express the answer\\
The required probability is \(\boxed{\frac{1}{4}}\) (or 0.25).
Hence, the probability that the music stops within the first half‑minute is \(\frac{1}{4}\).
Correct Answer: \(\frac{1}{4}\) (or 0.25)