Question

How many ways can you roll a pair of dice and get an even product?

Answer 1

Hint: This question belongs to the topic of probability. In this question, first we will get to know what will be the product of even and even, even and odd, odd and even, and odd and odd. After that, we will check how many ways there will be for rolling a pair of dice and get an odd product. After that, we will check the total number of odd and even products for rolling a pair of dice. And, from here we will get the number of ways for getting an even product.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to find the number of ways of getting an even product if we roll a pair of dice. It means if we roll two dice then what will be the possibility of getting an even product.
Let us first know what will be the product of odd and odd, even and odd, odd and even, and even and odd terms.
As we know that even term is always in the form of (2k) or (2k+2) and odd term is always in the form of (2k+1) or (2k-1), where k is an integer.
So,
The product of odd and odd will be \[\left( 2k+1 \right)\left( 2k-1 \right)=4{{k}^{2}}-1\]. This product is odd.
The product of odd and even or even and odd will be \[\left( 2k+1 \right)\left( 2k \right)=\left( 2k \right)\left( 2k+1 \right)=4{{k}^{2}} +1\]. This product is odd.
The product of even and even will be \[\left( 2k \right)\left( 2k \right)=4{{k}^{2}}\]. This product is even.
So, let us find out the probability for getting an odd product. As we know that for getting the product as odd from the two dices, we will have to take only odd numbers that are 1, 3, and 5.
So, for multiplying three numbers with the same three numbers we get combinations as \[3\times 3\] which is equal to 9.
Now, if we check for all the products for a pair of dice, then the combination will be \[6\times 6\] which is equal to 36.
As we can see that all the products will be either even or odd.
So, we can write for rolling a pair of dice
Total number of products getting either even or odd = Total number of products getting even + Total number of products getting odd
\[\Rightarrow \] 36= (Total number of products getting even) + (9)
Hence, the total number of products getting even will be 36-9 = 27.
So, by 27 ways we can roll a pair of dice and we will get an even product.

Note: We should have better knowledge in the chapter probability. We should know how many numbers are there in the dice for solving this type of question. We should know that the product of odd and odd is always an odd, product of odd and even or even and odd is always an odd, and also the product of even and even is always an even.