Probability of Drawing Odd Sum from a Set of Cards

In this article, we will delve into the probability of drawing two cards from a set of four, with numbered cards 1, 2, 3, and 4, such that the sum of the numbers on the drawn cards is odd. We will explore the theoretical calculations and provide a clear step-by-step breakdown of the solution.

Understanding the Concept

The sum of two integers is odd if one integer is odd and the other is even. This is a fundamental property of arithmetic. To solve the problem, we will use combinatorial methods and probability theory.

Step-by-Step Solution

Step 1: Identify the Total Number of Ways to Draw 2 Cards

The total number of ways to choose 2 cards out of 4 can be determined using the combination formula:

u232a4C2 u232a4u200b2 4! / (2! * (4 - 2)!) 6 ways

Step 2: Identify the Favorable Outcomes

To obtain an odd sum, the pair must consist of one odd and one even number. In our set, the odd numbers are 1 and 3, and the even numbers are 2 and 4. The possible pairs are:

1, 2 1, 4 3, 2 3, 4

These are the only pairs that result in an odd sum. Therefore, there are 4 favorable outcomes.

Step 3: Calculate the Probability

The probability that the sum of the numbers drawn is odd is given by the ratio of the number of favorable outcomes to the total number of outcomes:

P(sum is odd) 4 / 6 2 / 3

Conclusion

The probability that the sum of the numbers drawn is odd is 2 / 3.

Further Insight

Another approach to solving this problem is to consider that the first card drawn is irrelevant as long as the second card is chosen to ensure an odd sum. After drawing the first card, there are three remaining cards, two of which will form an odd sum, and one of which will form an even sum. Thus, the probability is 2 / 3.

Alternative Explanation

Alternatively, you can list all six pair combinations: 12, 13, 14, 23, 24, and 34. From these, the pairs that result in an even sum (12, 14, and 34) account for three scenarios, giving a probability of 3 / 6 1 / 2. Since the probability of an even sum is 1 / 2, the probability of an odd sum is 1 - 1 / 2 1 / 2 or 3 / 6 1 / 3, but since we are looking for the remaining probability, it is 2 / 3.

Similarly, the pairs that give an odd sum are 12, 34, 14, and 23, totaling four scenarios, thus the probability is 4 / 6 2 / 3.

Both approaches point to the same result, confirming that the probability of drawing an odd sum from the set of cards is indeed 2 / 3.