Question

In a school of 500 teens, 40 have taken drum lessons. What are the odds that a randomly chosen teen has not taken drum lessons?

Short answer

The odds that a randomly chosen teen has not taken drum lessons is 92%.

Step-by-step solution

Determine total number of teens

The problem mentions the total number of teens in the school is 500. This data is important to determine the percentage or odds of any event under consideration.

Identify number of teens who have taken drums lessons

The problem mentions that there are 40 teens who have taken drum lessons. This can be used to calculate the percentage of teens not taking drum lessons.

Calculate the number of teens who have not taken drum lessons

By subtracting the number of teenagers which have taken drum lessons from the total number of teenagers, this gives the number of teens who have not taken drum lessons. So, \( 500 - 40 = 460 \) students have not taken drum lesson.

Calculate the odds

The odds of an event is calculated by dividing the number of favorable instances by the total number of instances. Hence, the probability that a randomly chosen teen has not taken drum lessons is \( 460/500 \). This can be simplified via short division to 0.92 or converted to a percentage to be 92%.

Key concepts

Understanding Probability Theory

Probability theory is a branch of mathematics focused on analyzing random events and quantifying the likelihood of various outcomes. The core idea is that everything in the universe that happens by chance can be measured with numbers. When we speak of the 'odds' or 'probability' of an event, we're referring to a number between 0 and 1 (or 0% to 100%), where 0 indicates an impossibility and 1 (or 100%) denotes certainty.

For the case of our exercise, we are interested in finding the probability that a randomly chosen teen has not taken drum lessons. To do this, we must look at the ratio of the number of favorable outcomes (teens who haven't taken drum lessons) over the total number of possible outcomes (all teens). This approach helps us form a solid understanding of likelihood and can be applied to a vast array of scenarios in both everyday life and specialized fields such as science, finance, and more.

The Role of Complementary Events

Complementary events are two outcomes of an event that are the only possible outcomes and that cannot occur at the same time. They essentially 'complete' each other so that the sum of their probabilities adds up to 1 (or 100%). If we know the probability of an event occurring, we can easily find the probability of its complement (the event not occurring) by subtracting the known probability from 1.

In our exercise, the event is a teen having taken drum lessons, and its complement is a teen not having taken drum lessons. Given that we know 40 teens out of 500 have taken drum lessons, the probability of picking a teen at random who has taken drum lessons is \(\frac{40}{500}\) or 8%. Therefore, the complementary event – a teen not having taken drum lessons – would have a probability of 100% - 8%, which is 92%. Understanding complementary events is crucial as it simplifies calculations and enhances our intuitive grasp of probability.

Calculating Percentages

Percentage calculation is a mathematical concept used to express how large one quantity is, relative to another quantity. The term 'percent' comes from the Latin 'per centum,' meaning 'by the hundred.' When dealing with probabilities, we often convert the fraction or decimal to a percentage to make it easier to understand and compare.

Using the school exercise, once we know that 460 students have not taken drum lessons, we calculate this as a percentage of the total number of students to express the odds. We do that by dividing 460 by 500 to get \(\frac{460}{500} = 0.92\), and then multiplying by 100 to convert to a percentage, which gives us 92%. This percentage tells us that if we pick any student randomly, there's a 92% chance that the student has not taken drum lessons. Whether we're dealing with money, grades, or survey results, being adept with percentage calculations allows for a more comprehensive and practical interpretation of numbers and statistics in our daily lives.