Rajesh and Mahesh are playing a game. In this game, each player throws two dice and note down the numbers on the dice. By the rules of the game, Mahesh needs to get two numbers such that their product is a perfect square, in order to win the game. What is the probability that Mahesh will win the game?

Options

A. 0.11111111111111
B. 0.22222222222222
C. 0.33333333333333
D. 0.28571428571429

Answer

To solve the problem of finding the probability that Mahesh will win the game by rolling two dice and getting a product that is a perfect square, we can follow these steps:

### Step 1: Understand the Problem
Mahesh needs to roll two dice and the product of the numbers on the two dice must be a perfect square. A perfect square is a number that can be expressed as the square of an integer.

### Step 2: Identify the Possible Outcomes
When rolling two dice, each die has 6 faces. Therefore, the total number of outcomes when rolling two dice is:
\[ Total\ Outcomes = 6 \times 6 = 36 \]

### Step 3: Identify Perfect Squares
The perfect squares that can be formed from the products of the numbers on the dice are:
- \(1 = 1 \times 1\)
- \(4 = 2 \times 2\)
- \(9 = 3 \times 3\)
- \(16 = 4 \times 4\)
- \(25 = 5 \times 5\)
- \(36 = 6 \times 6\)

Additionally, products like \(1 \times 4\) or \(2 \times 2\) also yield perfect squares.

### Step 4: List Favorable Outcomes
Now we need to find all pairs of numbers (from 1 to 6) whose product is a perfect square. The pairs are:
- (1, 1)
- (1, 4)
- (2, 2)
- (3, 3)
- (4, 1)
- (4, 4)
- (5, 5)
- (6, 6)

We can also include pairs where the order matters, such as (1, 4) and (4, 1).

After listing all combinations, we find the following pairs yield perfect squares:
1. (1, 1)
2. (1, 4)
3. (4, 1)
4. (2, 2)
5. (3, 3)
6. (4, 4)
7. (5, 5)
8. (6, 6)

### Step 5: Count the Favorable Outcomes
From the pairs listed, we have:
- (1, 1)
- (1, 4)
- (4, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)

This gives us a total of **8 favorable outcomes**.

### Step 6: Calculate the Probability
The probability \(P\) that Mahesh will win the game is given by the formula:
\[ P = \frac{Number\ of\ Favorable\ Outcomes}{Total\ Outcomes} = \frac{8}{36} \]

### Step 7: Simplify the Probability
Now, we simplify the fraction:
\[ P = \frac{8}{36} = \frac{2}{9} \]

### Final Answer
The probability that Mahesh will win the game is:
\[ \frac{2}{9} \]

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