Expected Value: Is This Game Worth Playing?

What is expected value?

The expected value $E(X)$ is the long-run average outcome of a random process. It's what you'd average out to if you repeated the experiment infinitely many times.

$E(X) = \sum x_i \cdot p_i$

Multiply each possible outcome by its probability, then add them all up.

Step-by-step: Is this game worth playing?

Setup: Pay $5 to play. Win $20 with probability 0.2, win $0 with probability 0.8.

Step 1 — List all outcomes and probabilities:

• Win $20: probability $= 0.2$
• Win $0: probability $= 0.8$

Step 2 — Compute expected winnings:

$E(\text{winnings}) = 20 \times 0.2 + 0 \times 0.8 = 4 + 0 = \$4$

Step 3 — Subtract the cost:

$E(\text{net profit}) = \$4 - \$5 = -\$1$

Step 4 — Decision: On average, you lose $1 per game. The game is not worth playing.

Important: expected value ≠ any single outcome

You never actually lose exactly $1 in a single game — you either lose $5 (80% of the time) or gain $15 (20%). But averaged over many games, your per-game loss approaches $1.

After 1000 games: expected total loss $\approx$ $1000.

When is a game "fair"?

A game is fair when $E(\text{net}) = 0$ — neither player has an advantage. Casinos always design games with $E < 0$ for the player (the "house edge").

Try it in the visualization

Adjust the win amount, probability, and cost. The expected value updates. Simulate 1000 games to watch cumulative profit/loss converge to $E \times n$ — confirming the theoretical prediction.