Probability of Drawing Cards with One Number Being Twice the Other and the Sum Being Even

In this article, we explore a mathematical problem involving the probability of drawing two cards numbered from 1 to 50, where one number is exactly twice the other and the sum of these two numbers is even. This problem requires a step-by-step approach to identify the valid pairs and calculate the probability.

Step 1: Identifying Valid Pairs

Let's denote the two numbers as ( x ) and ( 2x ). For both ( x ) and ( 2x ) to be chosen from the set of cards numbered from 1 to 50, ( x ) must satisfy the condition:

( 2x leq 50 ) implies ( x leq 25 )

Hence, ( x ) can take values from 1 to 25. The valid pairs ( (x, 2x) ) are as follows:

(1, 2) (2, 4) (3, 6) (4, 8) (5, 10) (6, 12) (7, 14) (8, 16) (9, 18) (10, 20) (11, 22) (12, 24) (13, 26) (14, 28) (15, 30) (16, 32) (17, 34) (18, 36) (19, 38) (20, 40) (21, 42) (22, 44) (23, 46) (24, 48) (25, 50)

This gives us a total of 25 valid pairs ( (x, 2x) ).

Step 2: Ensuring the Sum is Even

Next, we need to check the condition that the sum ( x 2x 3x ) is even. For ( 3x ) to be even, ( x ) must be even. Therefore, the valid values for ( x ) are the even numbers from 1 to 25:

x 2 x 4 x 6 x 8 x 10 x 12 x 14 x 16 x 18 x 20 x 22 x 24

Thus, the valid pairs that meet both conditions are:

(2, 4) (4, 8) (6, 12) (8, 16) (10, 20) (12, 24) (14, 28) (16, 32) (18, 36) (20, 40) (22, 44) (24, 48)

This gives us a total of 12 valid pairs.

Step 3: Total Number of Ways to Choose 2 Cards

The total number of ways to choose 2 cards from 50 is given by the combination formula:

( binom{50}{2} frac{50 times 49}{2} 1225 )

Step 4: Calculating the Probability

The probability ( P ) that one number is twice the other and their sum is even is given by the ratio of the number of valid pairs to the total number of pairs:

( P frac{text{Number of valid pairs}}{text{Total pairs}} frac{12}{1225} )

Final Answer: The probability that one number is twice the other number and the sum of the two numbers is even is ( boxed{frac{12}{1225}} )

Key Takeaways

The probability of drawing a card where one number is twice the other is 25/50 or 1/2, and if you draw such a card, there is a 1/25 chance that the other card is twice that first card, giving a total chance of 25/50*1/25 1/50. All of the "other numbers" will be even, but only 12 of those 25 first numbers will also be even, and since we are adding an even number to an even number, only even numbers will produce an even sum. Thus, the probability is given by (12/25) * (1/50) 12/1250 or 6/625.