Question

From a collection of eight cards numbered 1 to 8, if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is:

• A. \( \frac{3}{14} \)
• B. \( \frac{6}{13} \)
• C. \( \frac{3}{16} \)
• D. \( \frac{1}{4} \)

Hint:

When determining the probability of a perfect square, first check the prime factorizations of all possible pairs, and then count the number of favorable pairs.

Answer 1

The Correct Option is C

To solve the problem, we need to determine the probability that the product of the numbers on two drawn cards is a perfect square.

Since there are 8 cards numbered from 1 to 8, and each one is drawn with replacement, the total number of possible outcomes when two cards are drawn is \(8 \times 8 = 64\).

Now, we will calculate the favorable outcomes where the product of the numbers is a perfect square:

• Perfect squares up to 64 include 1, 4, 9, 16, 25, 36, 49, 64.

Let's explore each perfect square:

• 1: Pairs \((1,1)\).
• 4: Pairs \((1,4), (4,1), (2,2)\).
• 9: Pairs \((1,9), (9,1), (3,3)\).
• 16: Pairs \((1,16), (16,1), (2,8), (8,2), (4,4)\).
• 36: Pairs \((1,36), (36,1), (6,6)\).

Note that pairs involving cards higher than 8 are not possible.

Counting the favorable outcomes above, the possibilities where the product is a perfect square are: \((1,1), (1,4), (4,1), (2,2), (1,9), (9,1), (3,3), (1,16), (16,1), (2,8), (8,2), (4,4)\).

Thus, there are a total of 12 favorable outcomes.

Finally, the probability of drawing two cards such that their product is a perfect square is:

\[\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{12}{64} = \frac{3}{16}\]

Therefore, the probability is \(\frac{3}{16}\).

The correct answer is \(\frac{3}{16}\).

Answer 2

To find the probability that the product of the numbers on the two drawn cards is a perfect square, we begin by understanding the problem. With replacement drawing means each card can appear in multiple events independently. This indicates that both cards drawn have equal probability for each draw. Let us solve this step by step:

Step 1: Total Possible Outcomes

Since each card can be numbered from 1 to 8, the total number of ways to draw two cards is:
8 (choices for the first card) × 8 (choices for the second card) = 64.

Step 2: Determine Perfect Squares

The product of the numbers is a perfect square if the result is a perfect square. Let's identify pairs \((a, b)\) such that \(a \times b\) is a perfect square:

• Pairs where both numbers are the same: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8). These account for 8 successful pairs.
• Pairs where numbers are different: We need perfect square products when \(a \neq b\). As the possible products can be 4, 9, 16, 25, 36, 49, 64, let's find relevant pairings:
  • (1,1) :\ 1
  • (2,2) :\ 4
  • (3,3) :\ 9
  • (4,4) :\ 16
  • (5,5) :\ 25
  • (6,6) :\ 36
  • (7,7) :\ 49
  • (8,8) :\ 64
  • Also valid are (1, 4) & (4, 1), (2, 8) & (8, 2), (4, 2) & (2, 4): Totaling 4 additional pairs.

Hence, successful cases = 8 (identical numbers) + 4 (additional squares) = 12.

Step 3: Calculate Probability

Probability of drawing two cards where the product is a perfect square = (Number of Successful Outcomes) / (Total Possible Outcomes) = \( \frac{12}{64} = \frac{3}{16} \). Therefore, the probability we are looking for is \( \frac{3}{16} \).

The correct answer is \( \frac{3}{16} \).