Probability Of Drawing An Odd Number Greater Than 10 From An Urn

Aug 7, 2025 by ADMIN 65 views

Probability of Drawing a Ball Greater Than 10 from an Urn of Odd-Numbered Balls

Hey guys! Let's dive into a probability problem that involves drawing balls from an urn. This is a classic scenario that helps us understand conditional probability. So, let's break it down step by step.

Problem Statement

We have an urn filled with balls numbered from 1 to 15. Each ball has an equal chance of being drawn. We pick one ball, and we know that the ball is odd-numbered. What is the probability that this ball is greater than 10?

Understanding the Problem

To tackle this probability question effectively, it's crucial to first grasp the concept of conditional probability. Conditional probability deals with the likelihood of an event occurring given that another event has already happened. In simpler terms, it's like saying, "What's the chance of this happening, knowing that this other thing is true?"

In our case, we're not just looking for the probability of drawing a ball greater than 10. We have an extra piece of information: the ball we drew is odd-numbered. This changes the landscape of our problem. Instead of considering all the balls from 1 to 15, we're now only focusing on the odd-numbered balls.

Think of it like this: imagine you have a bag of marbles, some red and some blue. If I ask you the probability of picking a red marble, you'd consider all the marbles in the bag. But if I tell you that you're only allowed to pick from the marbles that are also smooth, you'd only consider the smooth marbles when calculating the probability. The condition (being smooth) narrows down our possible outcomes.

So, to solve our urn problem, we need to consider the set of odd-numbered balls as our new "universe." We'll then determine how many of these odd-numbered balls are also greater than 10. This will give us the numerator for our probability calculation. The total number of odd-numbered balls will be our denominator. By dividing the number of favorable outcomes (odd balls greater than 10) by the total number of possible outcomes (all odd balls), we'll arrive at our conditional probability.

This approach highlights the importance of identifying and using the given condition to narrow down the sample space. It's a fundamental aspect of solving conditional probability problems, and it's what makes them so interesting and relevant in real-world applications, from medical diagnoses to financial risk assessment.

Identifying Odd Numbers

First, let's list the odd numbers between 1 and 15: 1, 3, 5, 7, 9, 11, 13, 15. There are 8 odd numbers in total. This forms our sample space, the set of possible outcomes given the condition that the ball is odd.

Identifying Odd Numbers Greater Than 10

Next, we need to find the odd numbers that are greater than 10. From our list of odd numbers, these are 11, 13, and 15. So, there are 3 odd numbers greater than 10. These are our favorable outcomes, the balls that meet both criteria: being odd and being greater than 10.

Calculating the Probability

Now, we can calculate the probability. The probability of drawing a ball greater than 10, given that it's odd, is the number of favorable outcomes divided by the total number of possible outcomes.

Favorable Outcomes (odd and greater than 10): 3 (11, 13, 15)
Total Possible Outcomes (odd numbers): 8 (1, 3, 5, 7, 9, 11, 13, 15)

So, the probability is 3/8.

To solidify our understanding, let's consider a different scenario. Suppose we had balls numbered from 1 to 20, and we knew the drawn ball was even. What would be the probability of it being less than 10? We'd follow the same steps: identify the even numbers, find the even numbers less than 10, and then calculate the probability.

Another way to think about it is using a Venn diagram. Imagine one circle representing odd numbers and another circle representing numbers greater than 10. The overlapping area represents the numbers that are both odd and greater than 10. The probability we're calculating is the ratio of the overlapping area to the entire "odd numbers" circle.

Understanding these different perspectives can help you tackle a wide range of probability problems with confidence. It's not just about memorizing formulas, but about visualizing the problem and understanding the underlying concepts. This makes problem-solving not only more effective but also more engaging and enjoyable!

Solution

The probability of drawing a ball greater than 10, given that it is odd, is 3/8. This isn't one of the options provided (A) 1/3. It's essential to perform the calculations accurately and compare the result with the given alternatives. Remember, in probability, it's crucial to consider all possibilities and use the information given to narrow down the options. Sometimes, the correct answer might not be explicitly listed, indicating a need to re-evaluate the problem or the choices provided. So, always double-check your work and think critically about the answer's reasonableness within the context of the problem.

Final Thoughts

Probability problems can be tricky, but with a systematic approach, they become much more manageable. Remember to always define your sample space, identify favorable outcomes, and then calculate the probability. Keep practicing, and you'll become a pro in no time!

I hope this explanation helps! Let me know if you have any other questions.