Huge Disparity
I randomly picked two distinct integers between 1 to 1000 inclusive. Is it more likely that the product of these two numbers to be odd or even?
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3 solutions
Possibilities:
1. O d d × O d d = O d d
2. E v e n × E v e n = E v e n
3. E v e n × O d d = E v e n
also,
4. O d d × E v e n = E v e n
The product will be even in the case of three possibilities. The probability of the product being an odd number is 4 1 , while the probability of it being even is 4 3 . All in all, it can be concluded that the product is more likely to be an even number than an odd number. Therefore, the answer is C h o i c e B .
The number-pair has 4 possibilities. even-even, odd-odd, odd-even, even-odd. But, the product basically has 3 possibilities. It's the product we are concerned with, not the possibilities of the pair of numbers.
I had earlier written 4 Possibilities. But, someone told me that it seems unreasonable. That's why, I edited the post.
I edited the post once again.
There is actually a 4th possibility which is O d d x E v e n = E v e n . Even though this is essentially the same thing as the 3rd possibility, the odds of each of these 4 choices are 4 1 . This means that the probability of the product being even is actually 4 3 , which still is greater than 2 1 .
There are 4 possibilities... Either the chosen number pairs are odd-odd, even-even, odd-even or even-odd.
Odd x even gives an even number. Odd x odd gives an odd number. Even x even gives an even number.
Therefore 3 out of 4 possible pair types give an even number so there's a 75% chance of an even product as opposed to a 25% chance of an odd product.
Almost complete. You need to show that each of these pairs are equally likely to occur (in terms of probability).
There are exactly the same number of odd and even numbers in the set from which we pick.
Therefore odd × odd = odd is as likely as even × even = even and they cancel out.
Since we have only cases of odd × even = even left, it must be more likely to get an even number as product of 2 distinct numbers which were picked from the given set.
Hence, the answer is It is more likely that the product is even.