Problem 7 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Twelve cards, numbered 1 to 12 , are placed in a box, mixed up throughly. Then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3 , then find the probability that it is an even number. [CBSE-2008]

Step-by-Step Solution

Answer

The probability is \( \frac{5}{9} \).

Step 1: Determine the Sample Space

First, we identify the total possible outcomes when a card is drawn from numbers 1 to 12. Since it is known that the number must be more than 3, the possible numbers are 4, 5, 6, 7, 8, 9, 10, 11, and 12, giving us a sample space of 9 numbers.

Step 2: Identify Favorable Outcomes

Next, we list the even numbers within our sample space (4, 5, 6, 7, 8, 9, 10, 11, 12). The even numbers are 4, 6, 8, 10, and 12, which totals to 5 possible favorable outcomes.

Step 3: Calculate the Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability that the card drawn is an even number is \( \frac{5}{9} \).

Key Concepts

Sample Space

When we discuss probability, one of the core concepts we deal with is the sample space. This refers to the set of all possible outcomes in a particular experiment or scenario. In our card drawing exercise, we begin with 12 cards, each numbered from 1 to 12. However, since it's known that the card drawn is more than 3, we narrow down this sample space. By eliminating cards 1, 2, and 3, our remaining possible outcomes are cards numbered 4 to 12. Hence, the sample space consists of these 9 outcomes:

• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12

Grasping the concept of sample space is crucial as it sets the stage for calculating probabilities. It defines the universe of possibilities within which favorable outcomes can exist.

Favorable Outcomes

Favorable outcomes are specific results within the sample space that we are interested in. In our scenario, we are looking to determine the probability that the card drawn is an even number. Identifying these favorable outcomes involves selecting only those numbers within the sample space that meet the desired criteria. From the sample space of cards numbered 4 to 12, the even numbers are the ones we're interested in, which are:

• 4
• 6
• 8
• 10
• 12

In total, there are 5 even numbers, which means there are 5 favorable outcomes. By understanding how to pinpoint favorable outcomes, we can efficiently calculate the probability of the desired event occurring.

Even Numbers

The essence of even numbers is that they are divisible by 2 without a remainder. In probability problems like our card drawing exercise, identifying even numbers is important as they form the basis for determining favorable outcomes. In our exercise, among the cards numbered from 4 to 12: 4, 6, 8, 10, and 12 are even. These numbers are part of the arithmetic sequence starting from 4 with a common difference of 2. Recognizing this pattern helps in quickly identifying even numbers in a structured set. Understanding the role of even numbers not only aids in calculating probabilities but also enhances number sense, allowing for quicker analyses and identifications in broader mathematical contexts.