Question

A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or \(5 ?\) (1) \(\frac{13}{25}\) (2) \(\frac{21}{50}\) (3) \(\frac{12}{25}\) (4) \(\frac{23}{50}\)

Short answer

Answer: (4) \(\frac{23}{50}\)

Step-by-step solution

Find the multiples of 3 and 5 in the given range.

We need to find the total number of multiples of 3 and 5 within the first 50 natural numbers. We can do this by dividing 50 by 3 and 5 and taking the integer part of the result. Multiples of 3: \(\left\lfloor\frac{50}{3}\right\rfloor = 16\) Multiples of 5: \(\left\lfloor\frac{50}{5}\right\rfloor = 10\)

Find the number of common multiples of 3 and 5

Now we need to find out how many numbers are multiples of both 3 and 5. This means we need to find the multiples of least common multiple (LCM) of 3 and 5 which is 15. Common multiples of 3 and 5: \(\left\lfloor\frac{50}{15}\right\rfloor = 3\)

Calculate the probability

To find the probability that a number is a multiple of 3 or 5, we need to find the sum of the multiples of 3 and 5, and then subtract the common multiples to avoid counting them twice. Total multiples of 3 or 5: \(16 + 10 - 3 = 23\) Now, we divide this number by the total number of natural numbers in the given range (i.e., 50) to find the probability: Probability = \(\frac{23}{50}\)

Match the probability with the given options

Now, compare the calculated probability with the given options: (1) \(\frac{13}{25}\) (2) \(\frac{21}{50}\) (3) \(\frac{12}{25}\) (4) \(\frac{23}{50}\) The correct answer is (4) \(\frac{23}{50}\).

Key concepts

Natural Numbers

Natural numbers are the most basic set of numbers used for counting. They start from 1 and go upwards without any end. These numbers do not include zero, fractions, or negative numbers. For instance, the set of first 50 natural numbers is 1, 2, 3, ..., up to 50.
Natural numbers are often symbolized by the letter \( N \). They are the foundation of number systems and are utilized in basic arithmetic operations like addition, subtraction, and multiplication.
Understanding natural numbers is essential in probability, as we often count the number of favorable outcomes from a set of possibilities.

Multiples

In mathematics, a multiple of a number is the product of that number and an integer. For example, the multiples of 3 include 3, 6, 9, 12, and so on.
When trying to find multiples within a given range, like from 1 to 50, we divide the largest number in the range (50) by the given number to find out how many multiples fit inside that range. For example, \( \left\lfloor \frac{50}{3} \right\rfloor = 16 \) tells us there are 16 multiples of 3 from 1 to 50. Similarly, \( \left\lfloor \frac{50}{5} \right\rfloor = 10 \) indicates 10 multiples of 5 in the same range.

Least Common Multiple (LCM)

The Least Common Multiple or LCM of two numbers is the smallest number that is a multiple of both numbers. It is found by listing out the multiples of each number and identifying the smallest multiple they share.
For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, ... and for 5 they are 5, 10, 15, 20, 25, .... The first common multiple is 15, hence the LCM of 3 and 5 is 15.
Finding the LCM is crucial when we want to count common outcomes, like common multiples of 3 and 5 within a range, without counting any value twice.

Counting Principles

Counting principles allow us to efficiently calculate possible outcomes. They form the basis for probability and combinatorics.
In probability problems, like finding the multiples of 3 or 5 in a range, we use counting principles to avoid double counting. First, we add together the number of each type of multiple, and then we subtract the count of any overlaps (using the LCM).
With the example in the exercise, you have:

• 16 multiples of 3
• 10 multiples of 5
• 3 multiples of both (LCM of 3 and 5)

Adding these gives us 16 + 10 = 26, but since 3 numbers are counted twice, we subtract them to get 23 unique multiples, applying the principles of counting accurately. This result is then divided by the total number of possible outcomes (50 natural numbers) to find the probability.