Probability of Drawing Balls Marked with Multiples of 5 or 7

In this article, we will explore the probability of drawing a ball marked with a number that is a multiple of 5 or 7 from a bag containing 20 balls marked 1 to 20. We will walk through the steps to solve similar problems and provide a detailed explanation.

Understanding the Problem

We start with a bag containing 20 balls, each marked with a unique number from 1 to 20. We randomly draw one ball and want to determine the probability that the number on the ball is a multiple of 5 or 7.

Step-by-Step Solution

Identifying Multiples

First, we identify the multiples of 5 and 7 within the range of 1 to 20.

Multiples of 5

The multiples of 5 within the range 1 to 20 are: 5, 10, 15, 20. This gives us a total of 4 numbers.

Multiples of 7

The multiples of 7 within the range 1 to 20 are: 7, 14. This gives us a total of 2 numbers.

Checking for Overlaps

Next, we need to check if there are any common multiples of 5 and 7 within the range of 1 to 20. The least common multiple (LCM) of 5 and 7 is 35, which is greater than 20. Therefore, there are no overlaps in this case.

Total Unique Favorable Outcomes

The total number of favorable outcomes is the sum of the multiples of 5 and multiples of 7:

[text{Total favorable outcomes} 4 text{ (multiples of 5)} 2 text{ (multiples of 7)} 6]

Total Possible Outcomes

The total number of possible outcomes is the total number of balls in the bag, which is 20.

Calculating the Probability

The probability (P) of drawing a ball that is a multiple of 5 or 7 is given by the ratio of favorable outcomes to total outcomes:

[P frac{6}{20} frac{3}{10}]

Therefore, the probability that a randomly drawn ball is marked with a number that is a multiple of 5 or 7 is (frac{3}{10}).

Generalizing the Solution

Example with 30 Balls

Consider a similar scenario where the bag contains 30 balls marked from 1 to 30. We want to find the probability of drawing a ball marked with a number that is a multiple of 5 or 7.

There are 6 multiples of 5: 5, 10, 15, 20, 25, 30. There are 4 multiples of 7: 7, 14, 21, 28. If we add these together, we have 10 numbers in total. However, we need to check for overlaps. The number 15 is a common multiple of both 5 and 7, so we subtract 1.

[text{Total favorable outcomes} 6 text{ (multiples of 5)} 4 text{ (multiples of 7)} - 1 text{ (overlap)} 9]

The total number of possible outcomes is 30. Therefore, the probability is:

[P frac{9}{30} frac{3}{10}]

Insights and Conclusion

In both scenarios, the probability of drawing a ball marked with a number that is a multiple of 5 or 7 is (frac{3}{10}). This demonstrates that the method of identifying multiples, checking for overlaps, and calculating the ratio of favorable to total outcomes is a consistent and reliable approach to solving probability problems involving multiples.

By understanding these steps, you can apply this method to similar problems, whether it involves 20 or 30 balls, or any other range of numbers. The key is to accurately identify the multiples and handle any overlaps to ensure an accurate calculation of the probability.