Two-Dice Product Probability

Two-dice product probability

When rolling two fair six-sided dice, each ordered pair ((1,1), (1,2), \dots, (6,6)) is equally likely. That gives 36 total outcomes. For a probability question about a product, the main task is to count how many pairs give the required product.

1) List the favorable pairs

For the target product, look for factor pairs that can come from two numbers between 1 and 6. Because the dice are labeled 1 through 6, only products that can be formed with those values matter.

2) Count outcomes carefully

If the two dice are different numbers, remember that order matters: ((2,3)) and ((3,2)) are two separate outcomes. If the numbers are the same, there is only one ordered pair such as ((4,4)).

3) Form the probability

Use [ P(\text{event})=\frac{\text{number of favorable outcomes}}{36}. ] Simplify the fraction if possible.

4) Check your work

Make sure every favorable pair really gives the stated product, and that you have not double-counted or missed any orderings. A quick sanity check is that the probability must be between 0 and 1.

This method works well for exact answers and helps avoid mistakes with repeated factors and ordering.