Question

1. two boxes each contain four tiles, numbered 1, 2, 3, and 4. the tiles are identical in shape and size. a student randomly draws one tile from each box and calculates the product of the two numbers on the tiles. which of the following products is most likely to occur? a 2 b 4 c 6 d 8

Explanation:

Step1: List all possible products

First, we consider the two boxes, each with tiles numbered 1, 2, 3, 4. When we draw one tile from each box, the possible pairs and their products are:

• \(1\times1 = 1\)
• \(1\times2 = 2\)
• \(1\times3 = 3\)
• \(1\times4 = 4\)
• \(2\times1 = 2\)
• \(2\times2 = 4\)
• \(2\times3 = 6\)
• \(2\times4 = 8\)
• \(3\times1 = 3\)
• \(3\times2 = 6\)
• \(3\times3 = 9\) (Wait, but the options are 2, 4, 6, 8. Wait, maybe the tiles are numbered 1, 2, 3, 4? Wait, the problem says "two boxes each contain four tiles numbered 1, 2, 3, and 4". So each box has 1,2,3,4. So the possible products:

Let's list all 16 possible outcomes (since each box has 4 tiles, 4x4=16 total possible draws):

From box 1 (1,2,3,4) and box 2 (1,2,3,4):

• (1,1): 1
• (1,2): 2
• (1,3): 3
• (1,4): 4
• (2,1): 2
• (2,2): 4
• (2,3): 6
• (2,4): 8
• (3,1): 3
• (3,2): 6
• (3,3): 9
• (3,4): 12
• (4,1): 4
• (4,2): 8
• (4,3): 12
• (4,4): 16

But the options are A.2, B.4, C.6, D.8. Wait, maybe I misread. Wait the options are A.2, B.4, C.6, D.8? Wait the original problem's options: A.2, B.4, C.6, D.8? Wait the image shows options A.2, B.4, C.6, D.8? Wait let's count the frequency of each product among the options:

• Product 2: occurs when (1,2) and (2,1) → 2 times
• Product 4: occurs when (1,4), (2,2), (4,1) → 3 times
• Product 6: occurs when (2,3), (3,2) → 2 times
• Product 8: occurs when (2,4), (4,2) → 2 times

Wait, wait, maybe the tiles are numbered 1,2,3,4? Wait no, maybe the first box has 1,2,3,4 and the second box too? Wait let's re - check:

Wait the problem says "two boxes each contain four tiles numbered 1, 2, 3, and 4". So each box has tiles 1,2,3,4. So the possible products for the options:

• For product 2: (1,2), (2,1) → 2 outcomes
• For product 4: (1,4), (2,2), (4,1) → 3 outcomes
• For product 6: (2,3), (3,2) → 2 outcomes
• For product 8: (2,4), (4,2) → 2 outcomes

Wait, but maybe I made a mistake. Wait, let's list all possible pairs again:

Box 1: [1,2,3,4], Box 2: [1,2,3,4]

All possible (a,b) where a ∈ Box1, b ∈ Box2:

(1,1):1, (1,2):2, (1,3):3, (1,4):4,

(2,1):2, (2,2):4, (2,3):6, (2,4):8,

(3,1):3, (3,2):6, (3,3):9, (3,4):12,

(4,1):4, (4,2):8, (4,3):12, (4,4):16.

Now, count the number of times each option product occurs:

• Product 2: (1,2), (2,1) → 2 times.

• Product 4: (1,4), (2,2), (4,1) → 3 times.

• Product 6: (2,3), (3,2) → 2 times.

• Product 8: (2,4), (4,2) → 2 times.

Since 3 > 2, the product 4 occurs most frequently. Wait, but wait, maybe the tiles are numbered differently? Wait, maybe the first box has 1,2,3,4 and the second box has 1,2,3,4? Wait, but according to the count, product 4 occurs 3 times, which is more than the others (2 times each for 2,6,8).

Wait, but maybe I misread the problem. Wait the problem says "two boxes each contain four tiles numbered 1, 2, 3, and 4". So each box has 1,2,3,4. So the calculation is as above.

Step2: Determine the most frequent product

We have the number of occurrences:

• Product 2: 2

• Product 4: 3

• Product 6: 2

• Product 8: 2

Since 3 is the highest frequency among these, the product 4 is the most likely to occur.

Answer:

B. 4