To find a suitable simulation design that Ashwin could use to estimate the probability that the next three songs are all pop songs, we need to ensure that the simulation reflects the equal likelihood of classical and pop songs.
Let's analyze each option in detail:
Option A (Random digits):
- Let [tex]\(1, 2, 3, 4, 5\)[/tex] represent classical songs.
- Let [tex]\(6, 7, 8, 9\)[/tex] represent pop songs.
- Select three random digits, then repeat the process.
Analysis:
In this option, there are 5 numbers representing classical songs and 4 numbers representing pop songs. Since the counts are not equal and we want to simulate an equal likelihood of selecting either type of song, this simulation design does not accurately reflect the equal probability scenario.
Option B (Random digits):
- Let [tex]\(1, 2, 3\)[/tex] represent classical songs.
- Let [tex]\(4, 5, 6\)[/tex] represent pop songs.
- Select three random digits, then repeat the process.
Analysis:
In this option, each type of song is represented by 3 digits, fitting the requirement of equal likelihood for each type. This simulation accurately reflects the equal probability scenario and is a suitable design.
Option C (Number cube):
- Let [tex]\(1\)[/tex] represent classical songs.
- Let [tex]\(2\)[/tex] represent pop songs.
- Roll the cube three times, then repeat the process.
Analysis:
Using a number cube (die), each outcome (number on the die) is equally likely. However, this option only uses two values (1 and 2), so it essentially becomes analogous to flipping a coin. Hence, this could be considered a valid method, but only if it were a cubic die where the numbers correspond correctly. Given it's a six-sided die translating to a binary outcome (1 or 2), this simplification works for equal probabilities.
Option D (Number cube):
- Let even number = classical songs.
- Let odd number = pop songs.
Analysis:
In this option, each roll of the die has equal likelihood of resulting in an even number (classical) or an odd number (pop). There are three even numbers (2, 4, 6) and three odd numbers (1, 3, 5), ensuring equal probability for both types of songs. This is a realistic and good simulation approach.
Conclusion:
The most suitable simulation designs are Option C (if clarified as a binary outcome) and Option D. However, if we consider common use and practicality in explaining simulation, Option D provides a clear representation using a number cube with even and odd outcomes.
Given the current options and analysis, Option D sounds simplest and thus could be the best approach:
D. Number cube: Let even number = classical, odd number = pop.
