What is uncertain?
A die has six possible outcomes. If it is fair, each has probability 1/6. Together those probabilities sum to one. A probability distribution describes the possible outcomes and how much probability each receives.
Change the distribution
In the experiment, choose the probability of rolling a 6. The remaining probability is shared equally among 1 through 5. Watch the bars and expected value change. This is a deliberately adjustable die, not a simulation of a particular physical die. Set P(6) to zero, then to one. The expected outcome shifts because the weights in the average have changed, even though no individual die roll is an average.
INTERACTIVE EXPLORATION
Build a distribution
Choose the probability of a 6; all remaining outcomes share the other probability equally. How does the expected value move?
What does the average mean?
The expected value is the sum of each outcome multiplied by its probability. For a fair die, it is 3.5. That does not mean the next roll can be 3.5; it is a long-run average under the stated distribution.
Check your understanding
Can a single roll of a standard six-sided die equal its expected value?
No. Its expected value is 3.5, but an individual roll is an integer from 1 through 6.