Probability of Two Even Numbers When Their Sum is Even Among the First 15 Natural Numbers

Understanding the probability of selecting two even numbers from the first 15 natural numbers, given that their sum is even, involves a detailed exploration of combinatorial methods. Let's delve into this problem step-by-step, using clear and structured reasoning to arrive at the final answer.

Step 1: Identify the First 15 Natural Numbers

The first 15 natural numbers are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Among these numbers, the even numbers are:

2, 4, 6, 8, 10, 12, 14

AND the odd numbers are:

1, 3, 5, 7, 9, 11, 13, 15

This breakdown will help us determine the total number of ways to choose the numbers.

Step 2: Determine the Total Ways to Choose Two Numbers

The total number of ways to choose 2 numbers from the first 15 natural numbers can be calculated using the combination formula:

(binom{n}{r} frac{n!}{r!(n-r)!})

For our case, we have:

(binom{15}{2} frac{15 times 14}{2 times 1} 105)

This is the total sample space.

Step 3: Identify Conditions for the Sum to be Even

The sum of two numbers is even if:

Both numbers are even (n1). Both numbers are odd (n2).

Let's calculate the number of favorable outcomes for each case.

Step 4: Calculate the Number of Favorable Outcomes

Case 1: Both Numbers are Even

There are 7 even numbers. The number of ways to select 2 even numbers is:

(binom{7}{2} frac{7 times 6}{2 times 1} 21)

Case 2: Both Numbers are Odd

There are 8 odd numbers. The number of ways to select 2 odd numbers is:

(binom{8}{2} frac{8 times 7}{2 times 1} 28)

Adding these cases together, the total number of ways to select two numbers such that their sum is even is:

21 (even even) 28 (odd odd) 49

Step 5: Calculate the Probability

The probability that two numbers selected are both even given that their sum is even is calculated as follows:

(P(text{both even} | text{sum is even}) frac{text{Number of ways to choose 2 even numbers}}{text{Total ways to choose 2 numbers with even sum}} frac{21}{49})

This simplifies to:

(P(text{both even} | text{sum is even}) frac{21}{49} frac{3}{7})

Thus, the probability that two numbers randomly selected from the first 15 natural numbers are both even given that their sum is even is:

(frac{3}{7})

Conclusion

This step-by-step approach helps us understand the detailed process of calculating the required probability. The combination formula and basic principles of probability play a crucial role in solving such problems.