In this lesson:
By the end of this lesson, you should be able to:
Last lesson: every distribution has a single balance point.
What one number summarizes where the whole distribution sits?
That number is the expected value. Let's compute it.
= number of heads, with :
A spinner has regions worth 1, 1, 1, 10. So , .
The plain average of is 5.5. Is that ?
Pick yes or no before advancing.
Don't average the values: the spinner's answer is 3.25, not 5.5
Don't divide by n: the weight is , which equals only if equally likely
Always multiply each value by its own probability.
You already computed a mean for data sets.
Expected value is the same operation — just with probabilities as the weights.
Let's see them side by side.
The expected value is the mean of the probability distribution, written .
40 trials: 10 zeros, 20 ones, 10 twos. Relative frequencies match the two-coin model.
A game has values 0, 5, 10, 20 with probabilities .
Find from scratch. Write each term as value × probability.
Commit to a number before advancing.
✓ — a weighted average
✓ Likelier values count more, not equally
✓ E(X) is the mean of the distribution, written
Next lesson, E(X) becomes a physical balance point under the histogram.
Then we add wins and losses — signed payoffs — to score real games.