AI Grounds
Open AI Grounds on a desktop
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Weight each payoff, then compare the average with the spread.
Two fictional bets with net payoffs. A random variable is the numerical payoff of one uncertain round. Change a probability or payoff to see its contribution to the average.
Chance of the positive payoff on each round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Net positive payoff on a winning round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Net negative payoff on a losing round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Win: 72%, +$35.00
Loss: 28%, −$18.00
EV = 0.72 × +$35.00 + 0.28 × (−$18.00)
+$25.20 + (−$5.04) = +$20.16
Chance of the positive payoff on each round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Net positive payoff on a winning round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Net negative payoff on a losing round. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Win: 34%, +$125.00
Loss: 66%, −$38.00
EV = 0.34 × +$125.00 + 0.66 × (−$38.00)
+$42.50 + (−$25.08) = +$17.42
| Bet | EV ($) | SD ($) | Loss chance | Break-even p |
|---|---|---|---|---|
| Bet A | +$20.16 | ≈ 23.80 | 28% | ≈ 34% |
| Bet B | +$17.42 | ≈ 77.21 | 66% | ≈ 23.3% |
Scroll the comparison table horizontally to see all columns.
Standard deviation (SD) measures the spread of one round, in dollars. Loss chance measures how often the negative payoff occurs. Break-even p is the win probability making EV zero.
Bet A has the higher EV. Bet A has the smaller per-round SD. SD is one aspect of risk, not a complete decision rule.
Variance = p × (win − EV)² + (1 − p) × (loss − EV)², in dollars squared. SD = √variance, in dollars. Break-even p = −loss / (win − loss), found by setting EV to zero.
Model average payoff per round.
+$20.16
p × win + (1 − p) × loss
Neither payoff needs to equal the average.
Model average payoff per round.
+$17.42
p × win + (1 − p) × loss
Neither payoff needs to equal the average.
Each bet has a seeded pseudorandom sample illustrating independent rounds with unchanged probabilities. Longer runs extend the same outcome sequence. Payoff edits keep wins and losses fixed; probability edits change which draws count as wins.
Solid: sample average. Dotted: model EV +$20.16. Vertical axis: average payoff in dollars; both charts use the same scale.
| Round | Outcome | Payoff ($) | Average ($) |
|---|---|---|---|
| 1 | Loss | −$18.00 | −$18.00 |
| 2 | Win | +$35.00 | +$8.50 |
| 3 | Win | +$35.00 | +$17.33 |
| 4 | Win | +$35.00 | +$21.75 |
| 5 | Loss | −$18.00 | +$13.80 |
| 6 | Win | +$35.00 | +$17.33 |
| 7 | Win | +$35.00 | +$19.86 |
| 8 | Loss | −$18.00 | +$15.13 |
| 9 | Win | +$35.00 | +$17.33 |
| 10 | Loss | −$18.00 | +$13.80 |
| 11 | Loss | −$18.00 | +$10.91 |
| 12 | Win | +$35.00 | +$12.92 |
| 13 | Win | +$35.00 | +$14.62 |
| 14 | Loss | −$18.00 | +$12.29 |
| 15 | Loss | −$18.00 | +$10.27 |
| 16 | Win | +$35.00 | +$11.81 |
| 17 | Win | +$35.00 | +$13.18 |
| 18 | Win | +$35.00 | +$14.39 |
| 19 | Win | +$35.00 | +$15.47 |
| 20 | Win | +$35.00 | +$16.45 |
| 21 | Win | +$35.00 | +$17.33 |
| 22 | Win | +$35.00 | +$18.14 |
| 23 | Loss | −$18.00 | +$16.57 |
| 24 | Win | +$35.00 | +$17.33 |
| 25 | Win | +$35.00 | +$18.04 |
| 26 | Win | +$35.00 | +$18.69 |
| 27 | Win | +$35.00 | +$19.30 |
| 28 | Win | +$35.00 | +$19.86 |
| 29 | Win | +$35.00 | +$20.38 |
| 30 | Win | +$35.00 | +$20.87 |
| 31 | Win | +$35.00 | +$21.32 |
| 32 | Win | +$35.00 | +$21.75 |
| 33 | Win | +$35.00 | +$22.15 |
| 34 | Win | +$35.00 | +$22.53 |
| 35 | Win | +$35.00 | +$22.89 |
| 36 | Win | +$35.00 | +$23.22 |
| 37 | Loss | −$18.00 | +$22.11 |
| 38 | Win | +$35.00 | +$22.45 |
| 39 | Win | +$35.00 | +$22.77 |
| 40 | Win | +$35.00 | +$23.07 |
| 41 | Win | +$35.00 | +$23.37 |
| 42 | Win | +$35.00 | +$23.64 |
| 43 | Win | +$35.00 | +$23.91 |
| 44 | Win | +$35.00 | +$24.16 |
| 45 | Win | +$35.00 | +$24.40 |
| 46 | Loss | −$18.00 | +$23.48 |
| 47 | Loss | −$18.00 | +$22.60 |
| 48 | Win | +$35.00 | +$22.85 |
| 49 | Win | +$35.00 | +$23.10 |
| 50 | Win | +$35.00 | +$23.34 |
| 51 | Loss | −$18.00 | +$22.53 |
| 52 | Win | +$35.00 | +$22.77 |
| 53 | Loss | −$18.00 | +$22.00 |
| 54 | Win | +$35.00 | +$22.24 |
| 55 | Loss | −$18.00 | +$21.51 |
| 56 | Loss | −$18.00 | +$20.80 |
| 57 | Loss | −$18.00 | +$20.12 |
| 58 | Win | +$35.00 | +$20.38 |
| 59 | Loss | −$18.00 | +$19.73 |
| 60 | Loss | −$18.00 | +$19.10 |
Solid: sample average. Dotted: model EV +$17.42. Vertical axis: average payoff in dollars; both charts use the same scale.
| Round | Outcome | Payoff ($) | Average ($) |
|---|---|---|---|
| 1 | Loss | −$38.00 | −$38.00 |
| 2 | Win | +$125.00 | +$43.50 |
| 3 | Loss | −$38.00 | +$16.33 |
| 4 | Win | +$125.00 | +$43.50 |
| 5 | Win | +$125.00 | +$59.80 |
| 6 | Loss | −$38.00 | +$43.50 |
| 7 | Win | +$125.00 | +$55.14 |
| 8 | Loss | −$38.00 | +$43.50 |
| 9 | Loss | −$38.00 | +$34.44 |
| 10 | Win | +$125.00 | +$43.50 |
| 11 | Win | +$125.00 | +$50.91 |
| 12 | Loss | −$38.00 | +$43.50 |
| 13 | Loss | −$38.00 | +$37.23 |
| 14 | Win | +$125.00 | +$43.50 |
| 15 | Loss | −$38.00 | +$38.07 |
| 16 | Loss | −$38.00 | +$33.31 |
| 17 | Loss | −$38.00 | +$29.12 |
| 18 | Win | +$125.00 | +$34.44 |
| 19 | Loss | −$38.00 | +$30.63 |
| 20 | Loss | −$38.00 | +$27.20 |
| 21 | Loss | −$38.00 | +$24.10 |
| 22 | Loss | −$38.00 | +$21.27 |
| 23 | Loss | −$38.00 | +$18.70 |
| 24 | Loss | −$38.00 | +$16.33 |
| 25 | Loss | −$38.00 | +$14.16 |
| 26 | Loss | −$38.00 | +$12.15 |
| 27 | Loss | −$38.00 | +$10.30 |
| 28 | Loss | −$38.00 | +$8.57 |
| 29 | Loss | −$38.00 | +$6.97 |
| 30 | Loss | −$38.00 | +$5.47 |
| 31 | Loss | −$38.00 | +$4.06 |
| 32 | Loss | −$38.00 | +$2.75 |
| 33 | Loss | −$38.00 | +$1.52 |
| 34 | Loss | −$38.00 | +$0.35 |
| 35 | Loss | −$38.00 | −$0.74 |
| 36 | Win | +$125.00 | +$2.75 |
| 37 | Win | +$125.00 | +$6.05 |
| 38 | Loss | −$38.00 | +$4.89 |
| 39 | Win | +$125.00 | +$7.97 |
| 40 | Loss | −$38.00 | +$6.83 |
| 41 | Win | +$125.00 | +$9.71 |
| 42 | Loss | −$38.00 | +$8.57 |
| 43 | Loss | −$38.00 | +$7.49 |
| 44 | Win | +$125.00 | +$10.16 |
| 45 | Loss | −$38.00 | +$9.09 |
| 46 | Loss | −$38.00 | +$8.07 |
| 47 | Win | +$125.00 | +$10.55 |
| 48 | Win | +$125.00 | +$12.94 |
| 49 | Win | +$125.00 | +$15.22 |
| 50 | Loss | −$38.00 | +$14.16 |
| 51 | Loss | −$38.00 | +$13.14 |
| 52 | Loss | −$38.00 | +$12.15 |
| 53 | Win | +$125.00 | +$14.28 |
| 54 | Win | +$125.00 | +$16.33 |
| 55 | Loss | −$38.00 | +$15.35 |
| 56 | Loss | −$38.00 | +$14.39 |
| 57 | Win | +$125.00 | +$16.33 |
| 58 | Win | +$125.00 | +$18.21 |
| 59 | Loss | −$38.00 | +$17.25 |
| 60 | Loss | −$38.00 | +$16.33 |
Under independent fixed-probability repeats, the sample average approaches EV in the long run. A finite run need not match it exactly, and adding rounds need not move closer every time.
Bet A: win 72%, EV +$20.16, SD about 23.80 dollars, 60-round sample average +$19.10. Bet B: win 34%, EV +$17.42, SD about 77.21 dollars, 60-round sample average +$16.33.