1. A record label held auditions and suitable candidates were invited for assessment in beat making and singing (vocalists). 23 candidates attended the auditions. - 4 people make beats only. - people are both vocalists and beat makers. A) Draw a Venn diagram to represent the above information, showing all possible outcomes. A candidate is selected at random: B) Determine the probability that it will be a beat maker. C) Determine the probability that it will be a beat maker but not a vocalist. D) Determine the probability that it will be a vocalist and not a beat maker. E) What is the probability that it will be either a beat maker or a vocalist? Express your answer as a percentage. F) Determine the probability that it will be a vocalist. G) Determine the probability that the candidate is a vocalist and also makes beats. H) Are the events "the candidate is a vocalist" and "the candidate is a beat maker" exhaustive events? Elaborate.
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To solve the problem, let’s break down the information given: 1. There are 23 candidates in total. 2. Four candidates only make beats, leaving us with 19 who are either vocalists or both vocalists and beat makers. 3. Let's denote: - V = number of vocalists - B = number of beat makers - VB = number of candidates who are both vocalists and beat makers From our breakdown: - **Pure Beat Makers** (B only) = 4 - **Total Beat Makers (B)** = 4 + VB - **Total Vocalists (V)** = VB + (number of pure vocalists) - **Total Candidates** = 23 Assuming there are x candidates who are pure vocalists (Vonly): We can set up the equation: \[ 4 + VB + (VB + x) = 23 \] which simplifies to: \[ 4 + 2VB + x = 23 \] From this equation, various values for VB will yield different distributions of the candidates. Now we can answer parts B to H. However, to answer these parts accurately, we need more information, specifically about how many candidates are only vocalists (Vonly). Assuming there’s one candidate who is both a vocalist and a beat maker, the breakdown is as follows: - V = 19 (total without pure beat makers) - VB = 1 (assuming 1 candidate is both) - Therefore, x (pure vocalists) = V - VB = 18 ### A) Venn Diagram Representation Create two overlapping circles. The left circle represents beat makers, and the right circle represents vocalists. The intersection is where candidates who are both beat makers and vocalists reside. ### B) Probability of a Beat Maker The total number of beat makers = 4 (only beat makers) + 1 (both) = 5. So, the probability: \[ P(B) = \frac{5}{23} \] ### C) Probability of a Beat Maker but not a Vocalist Only beat makers = 4; thus, \[ P(B \text{ only}) = \frac{4}{23} \] ### D) Probability of a Vocalist and not a Beat Maker If there are 18 vocalists (Vonly), we use: \[ P(V \text{ only}) = \frac{18}{23} \] ### E) Probability of Being Either a Beat Maker or a Vocalist Total candidates either as B or V = 23 (by the setup). Thus: \[ P(B \text{ or } V) = \frac{19}{23} \text{ or } 82.61\% \] ### F) Probability of a Vocalist \[ P(V) = \frac{19}{23} \] ### G) Probability that the Candidate is Vocalist and also makes Beats Assuming VB = 1: \[ P(V \text{ and } B) = \frac{1}{23} \] ### H) Are the events exhaustive? No, as it doesn't consider candidates who neither make beats nor sing, indicating that not all possible outcomes of candidate classification are covered. These probabilities illustrate the intersections and distinctions among the different candidate roles within the audition context. It’s like figuring out the perfect blend in music – sometimes you need a solo artist, and at other times, a collaboration shines!
