The learner determines theoretical probabilities for outcomes of one or two fair six-sided dice by identifying the equally likely sample space and expressing the ratio of favorable outcomes to total outcomes as a fraction, decimal, or percent. For two dice, this includes recognizing that ordered pairs provide 36 equally likely outcomes and that sums are not equally likely because different numbers of pairs produce them; conditional probability and more advanced probability distributions are not included.
For a probability with dice, use
Example: What is the probability that the sum of two fair six-sided dice is ?
Step 1: Count the total outcomes.
Each die has possible results. For two dice, the outcomes are ordered pairs, such as , meaning the first die shows and the second shows .
So there are
equally likely outcomes.
Step 2: Find the favorable outcomes.
The pairs that add to are
There are favorable outcomes.
Step 3: Write and simplify the probability.
As a decimal and percent:
So, the probability of rolling a sum of is
Remember: With two dice, there are equally likely ordered pairs. The sums themselves are not equally likely because some sums can be made in more ways than others.
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