Understanding Probability with Instrument Playing

In this article, we will explore a probability problem involving a group of 39 pupils who play musical instruments. The objective is to understand the probability of a randomly selected student playing both the flute and the piano. Let's break down the problem with detailed steps and reasoning.

Problem Scenario

Consider a group of 39 pupils. Some details are given about their instrument playing habits:

10 pupils play the flute only. 1 pupil plays the piano only. 13 pupils play neither instrument. We need to find the probability that a randomly selected student plays both the flute and the piano.

Analysis and Steps

First, let's summarize the given information:

Total number of pupils 39 Number of pupils who play the flute only 10 Number of pupils who play the piano only 1 Number of pupils who play neither instrument 13

When we sum up the pupils who play either of the instruments or neither, we obtain:

Common Logical Approach

Number of pupils who play either the flute or the piano or neither 10 (only flute) 1 (only piano) 13 (neither) 24 Number of pupils who play both 39 (total) - 24 (either or neither) 15 Probability that a randomly selected student plays both instruments 15/39 5/13 ≈ 38.5%

Another Approach with Venn Diagram

Visualizing the problem using a Venn diagram can also clarify the logic:

Draw two overlapping circles, one for the flute players and one for the piano players. Fill in the appropriate numbers: 10 in the flute only section. 1 in the piano only section. 13 outside the circles (neither). The overlapping area (both) is calculated as: 39 - 24 (neither or neither) 15. Thus, the probability that a randomly selected student plays both instruments is: 15/39 5/13 ≈ 38.5%.

Conclusion

In this problem, the key is to carefully analyze the given data and apply basic probability principles. The probability of a randomly selected student playing both the flute and the piano is 5/13, or approximately 38.5%.

For more probability and mathematics queries or any other related topics, feel free to explore my Quora Profile. It's a great resource for diving deeper into these topics and understanding complex concepts.