Question
Polly was a spectator at her town's air guitar competition. Contestants were allowed to play either the acoustic or electric air guitar, but not both. Polly recorded which type of guitar each contestant played. She also counted the number of contestants wearing different kinds of pants, as there were some interesting stylistic choices.
Acoustic guitar Electric guitar
Leather 5 3
Velvet 5 3
What is the probability that a randomly selected contestant was dressed in velvet and played an acoustic guitar?
Simplify any fractions.
Answers
GPT-4o mini
To find the probability that a randomly selected contestant was dressed in velvet and played an acoustic guitar, we first need to identify the total number of contestants and the number of contestants who fit the criteria of being dressed in velvet and playing an acoustic guitar.
From the data provided:
- Number of contestants who played acoustic guitar:
- Leather: 5
- Velvet: 5
- Total for acoustic: \(5 + 5 = 10\)
- Number of contestants who played electric guitar:
- Leather: 3
- Velvet: 3
- Total for electric: \(3 + 3 = 6\)
Now, we can calculate the total number of contestants:
\[
\text{Total contestants} = \text{Total for acoustic} + \text{Total for electric} = 10 + 6 = 16
\]
Next, we focus on the number of contestants who were dressed in velvet and played an acoustic guitar. According to the data:
- Number of contestants in velvet who played acoustic guitar = 5
Now, the probability \( P \) that a randomly selected contestant was dressed in velvet and played an acoustic guitar is given by the formula:
\[
P(\text{Velvet and Acoustic}) = \frac{\text{Number of Velvet and Acoustic}}{\text{Total Contestants}} = \frac{5}{16}
\]
Thus, the probability that a randomly selected contestant was dressed in velvet and played an acoustic guitar is:
\[
\boxed{\frac{5}{16}}
\]
From the data provided:
- Number of contestants who played acoustic guitar:
- Leather: 5
- Velvet: 5
- Total for acoustic: \(5 + 5 = 10\)
- Number of contestants who played electric guitar:
- Leather: 3
- Velvet: 3
- Total for electric: \(3 + 3 = 6\)
Now, we can calculate the total number of contestants:
\[
\text{Total contestants} = \text{Total for acoustic} + \text{Total for electric} = 10 + 6 = 16
\]
Next, we focus on the number of contestants who were dressed in velvet and played an acoustic guitar. According to the data:
- Number of contestants in velvet who played acoustic guitar = 5
Now, the probability \( P \) that a randomly selected contestant was dressed in velvet and played an acoustic guitar is given by the formula:
\[
P(\text{Velvet and Acoustic}) = \frac{\text{Number of Velvet and Acoustic}}{\text{Total Contestants}} = \frac{5}{16}
\]
Thus, the probability that a randomly selected contestant was dressed in velvet and played an acoustic guitar is:
\[
\boxed{\frac{5}{16}}
\]