Problem 12 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A bag contains 30 balls numbered from 1 to 30; 1 ball is drawn randomly. The probability that the number of the ball is multiple of 5 or 7 is (a) \(1 / 2\) (b) \(1 / 3\) (c) \(2 / 3\) (d) \(1 / 4\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{3} \).

Step 1: Understand the Problem

We need to find the probability of drawing a ball that is a multiple of 5 or 7 from a bag containing balls numbered from 1 to 30. This involves calculating how many such balls exist within the set of 30.

Step 2: Identify Multiples of 5

List all numbers from 1 to 30 that are multiples of 5. These numbers are: 5, 10, 15, 20, 25, and 30. There are 6 multiples of 5.

Step 3: Identify Multiples of 7

List all numbers from 1 to 30 that are multiples of 7. These numbers are: 7, 14, 21, and 28. There are 4 multiples of 7.

Step 4: Check for Overlapping Multiples

Identify any numbers that are multiples of both 5 and 7 by finding multiples of 35 within 30 (none in this case since 35 > 30). Thus, there is no overlap.

Step 5: Apply the Addition Rule for Probability

Since there are no overlapping numbers (numbers that are multiples of both 5 and 7), we can simply add the counts from Step 2 and Step 3: 6 multiples of 5 + 4 multiples of 7 = 10 numbers.

Step 6: Calculate the Probability

The probability is the number of favorable outcomes (10) over the total possible outcomes (30). Therefore, the probability is \( \frac{10}{30} = \frac{1}{3} \).

Key Concepts

Multiples of Numbers

Understanding multiples of numbers is crucial when solving probability problems like the one in the exercise. A multiple of a number is the result of that number multiplied by an integer. For instance, multiples of 5 include 5, 10, 15, and so on, because these are the products yielded from multiplying 5 by 1, 2, 3, etc.

In the exercise, we identified multiples of 5 and 7 within the range of 1 to 30:

• Multiples of 5: 5, 10, 15, 20, 25, and 30. Notice how each of these numbers can be divided by 5 without a remainder.
• Multiples of 7: 7, 14, 21, and 28. Just as with 5, these numbers can be divided by 7 cleanly.

When dealing with probability, recognizing these multiples allows us to define the set of numbers that meet the criteria of the question. Thus, the key is listing these multiples correctly to proceed with probability calculations.

Addition Rule for Probability

The Addition Rule for Probability is a helpful framework when calculating the likelihood of either of two events occurring, especially when the events are mutually exclusive - meaning they can't both occur at the same time.

In the exercise's context, picking a number that is a multiple of either 5 or 7 involves using this rule. The rule states:

• If sets A and B are mutually exclusive, the probability of A or B occurring is simply the sum of their individual probabilities.

Since there are no numbers between 1 and 30 that are multiples of both 5 and 7 (i.e., they don't overlap), we treat these sets as mutually exclusive. By applying the addition rule:

• Number of favorable outcomes ( either multiple of 5 or 7): 6 + 4 = 10.
• Total possible outcomes: 30.

This formula guides us to a neat solution and shows how addition is used to handle simpler probabilities before combining them.

Counting Outcomes

Understanding how to count outcomes is foundational in probability calculations. Outcomes are the results that can occur from a probability experiment. When our problem asked us to figure out the likelihood of drawing a ball numbered as a multiple of either 5 or 7, counting becomes essential.

In this exercise:

• The total number of balls, or possible outcomes, is 30.
• Using our earlier multiple identifications, we found: 6 multiples of 5 and 4 multiples of 7.
• Since there were no overlaps (numbers multiple of both 5 and 7), the count of favorable outcomes was simply 10.

Counting helps to establish both the numerator and the denominator of the probability fraction, which is essential in calculating probabilities accurately. Through this method, we are able to successfully determine that the probability of drawing a ball that is either a multiple of 5 or 7 is \( \frac{10}{30} = \frac{1}{3} \).