Balance Point and Signed Payoffs

Lesson 2 of 2: Picturing E(X) and Scoring Games

In this lesson:

• See expected value as a balance point
• Read E(X) as a long-run average
• Score games with wins and losses

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Locate expected value as the balance point of the histogram
2. Interpret expected value as the long-run average over many trials
3. Compute E(X) for signed payoffs and classify a game as fair, favorable, or unfavorable

Where Does E(X) Sit on the Graph?

We computed for the two coins.

It's a number — but where is it on the histogram?

Last lesson called it a balance point. Let's make that literal.

The Fulcrum Balances at One

For a symmetric distribution, the balance point is at the center.

Skew Pulls the Balance Point

The 1,1,1,10 spinner balances at 3.25 — pulled toward the long tail.

Predict: Can E(X) Equal 1.5 Girls?

Three children, = number of girls. The computed .

But no family has 1.5 girls. Can that be right?

Decide before advancing.

A Balance Point Sits Between the Bars

is correct — a fulcrum balances between the bars.

• Like "the average household has 2.3 people" — no household has 2.3
• E(X) is the long-run average, not a single result

Your Turn: Predict the Balance Point

Here's a histogram skewed to the left (tall bars on the high values).

Before computing — where does the fulcrum sit: low, center, or high?

Predict from the shape, then we'll check.

From Balance Point to Long-Run Average

A balance point is a static picture. There's also a dynamic meaning:

Repeat the situation many times — what's the average result?

The Running Average Settles to E(X)

Toss two coins thousands of times: the average number of heads settles to 1.

Payoffs Can Be Negative Losses

When the random variable is a payoff, its values can be losses.

• A win is a positive net change
• A loss is a negative one — the formula handles signs directly

Build the Signed Payoff Table

Pay $1 to play. Win $3 with probability , else lose your dollar.

Compute and Interpret the Payoff

You expect to lose 25 cents per play, on average.

Fair, Favorable, or Unfavorable to You

• → fair game
• → favorable to the player
• → unfavorable (favors the house)

Our game: , so it favors the house.

Your Turn: Classify Three Games

Game A: . Game B: . Game C: .

Classify each, and rank them best to worst for the player.

Commit before advancing.

Two Payoff Traps to Watch For

E(X) can be unattainable — 1.5 girls is correct; an average lands between values

Encode losses as negative — an unfavorable game must give

What This Lesson Gave You

✓ E(X) is the balance point of the histogram

✓ It's the long-run average over many trials

✓ For payoffs, its sign classifies the game

Coming Up Next: Building Distributions

You can now define, graph, and summarize a distribution with E(X).

Next standards: build distributions two ways — from a model and from data — then score real decisions.