Question

A three digit number is to be formed using the digits \(3,4,7,8\) and 2 without repetition. What is the probability that it is an odd number? (1) \(\frac{2}{5}\) (2) \(\frac{1}{5}\) (3) \(\frac{4}{5}\) (4) \(\frac{3}{5}\)

Short answer

Answer: (2) \(\frac{1}{5}\)

Step-by-step solution

Find the total possible three-digit numbers

To find the total possible three-digit numbers without repetition, we'll have 5 choices for the first digit (3, 4, 7, 8, and 2), 4 choices for the second digit (one less than previously since we can't repeat the first digit), and 3 choices for the third digit (we can't repeat either of the first two digits). Thus, the total possible three-digit numbers without repetition are: \(5\times4\times3=60\)

Find the total possible odd three-digit numbers

For a three-digit number to be odd, its last digit must be an odd number. In our set, we have only two odd digits (3 and 7). The first digit can’t be 2 since it has to be a three-digit number. Thus, we have \(3\) choices for the first digit (3, 4, and 8) and \(2\) choices for the last digit (3 and 7). For the second digit, we have \(2\) choices (the remaining two numbers after selecting the first digit). Thus, the total possible odd three-digit numbers are: \(3\times2\times2=12\)

Calculate the probability of forming an odd three-digit number

Now, we can calculate the probability of forming an odd three-digit number by dividing the number of odd three-digit numbers by the total possible three-digit numbers: \(\frac{12}{60} = \frac{1}{5}\) So the probability that a three-digit number formed from the given digits without repetition is an odd number is \(\frac{1}{5}\). The correct answer is (2) \(\frac{1}{5}\).

Key concepts

Understanding Three-Digit Numbers

When learning about three-digit numbers, we’re referring to numbers that range from 100 to 999. These numbers are made up of three place values: the hundreds, tens, and ones. For instance, in the number 347, '3' represents the hundreds place, '4' represents the tens place, and '7' is in the ones place.

In our particular problem, we're tasked with creating a three-digit number from a set of given digits: 3, 4, 7, 8, and 2. It's important to remember that each place value must be occupied by a single digit, and we cannot use the same digit twice in the same number if we are to follow the 'without repetition' rule. Hence, when creating a three-digit number, the choice for each subsequent digit becomes limited as the previous digits are chosen.

Permutations Without Repetition

The concept of permutations without repetition is central to many probability problems. A permutation is an arrangement of objects in a specific order, and when we say 'without repetition,' it means each object can be used only once.

Let's dive a bit deeper using our example: For the first digit of the three-digit number, we have five choices. Once a digit is picked for the hundreds place, it cannot be used again. Thus, we only have four choices for the tens place. This pattern continues, leaving us with three choices for the ones place. The mathematical way to express the total number of permutations in this case is by multiplying the number of choices at each step:

• First digit: 5 choices
• Second digit: 4 choices
• Third digit: 3 choices

So, we calculate the total permutations as: \(5 \times 4 \times 3 = 60\) permutations.

Calculating Odd Number Probability

Probability questions like this one often require us to determine the likelihood of a specific condition being met—in this case, that the three-digit number formed is odd. An odd number is characterized by having 1, 3, 5, 7, or 9 in the ones place. Considering our given digits, only 3 and 7 will make a number odd.

How do we find the probability? We determine the total number of odd numbers possible and divide that by the total number of three-digit numbers from the step discussing permutations. As worked out in the solution, there are 12 possible odd three-digit numbers and 60 total permutations. The probability is then calculated as \(\frac{12}{60} = \frac{1}{5}\), indicating there's a 20% chance of the three-digit number being odd.

Probability problems often involve this kind of logical breakdown: identifying the total set of outcomes and then finding the subset which fulfills our specific criteria.