To solve this problem, we can use the sum of probabilities rule, which states that the probability of two independent events occurring together is equal to the product of their individual probabilities.
To find the probability that the sum of the two numbers is less than zero, we need to consider all the possible pairs of numbers that could be selected. For each number, there are four possible pairs that could be formed by selecting another number from the bag. We can represent all the possible pairs in a table like this:
| | -4 | -2 | -1 | 2 |
|---|----|----|----|----|
| -4| -8 | -6 | -5 | -2 |
| -2| -6 | -4 | -3 | 0 |
| -1| -5 | -3 | -2 | 1 |
| 2| -2 | 0 | 1 | 4 |
In this table, each cell represents the sum of the two numbers in that row and column. For example, the cell in the first row and first column represents the sum of -4 and -4, which is -8.
To find the probability that the sum of the two numbers is less than zero, we need to count the number of pairs that have a negative sum and divide by the total number of possible pairs. From the table, we can see that there are 6 pairs that have a negative sum: (-4, -2), (-4, -1), (-2, -4), (-2, -1), (-1, -4), and (-1, -2). The total number of possible pairs is 4 x 4 = 16, since there are four numbers and we are selecting two with replacement.
Therefore, the probability that the sum of the two numbers is less than zero is 6/16, which simplifies to 3/8. So the answer is 3/8.