doing math

courses › Thinking

Risk & the long run

level 35 course

Volatility, losing streaks, spreading bets, and outliers.

In your head, jot if needed · no calculator why?

Learn first (about 3 minutes)

Opens at level 35.

Sign in to start

Builds on: Base rates (not open yet) · Growth & projection (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Averages hide risk. Four facts show where.

A gain and an equal percent loss don't cancel: +50% then −50% turns $100 into $75. The chance of a run of independent losses shrinks fast, because its chances multiply. One huge value drags the mean, the total divided by the count, but barely moves the median, the middle of the sorted values. And across n independent, identical bets, the typical swing of the total, its standard deviation, grows by √n, not n, so the average per bet steadies.

Techniques

Multiply the changes

A percent gain and a percent loss, one after the other.

  1. Turn each percent into a multiplier: +25% is × 1.25, −25% is × 0.75.
  2. Apply them in order.
  3. Equal ups and downs always end below the start.
worked example

Example: $800 gains 25%, then loses 25%. What is it worth now?

  1. $800 × 1.25 = $1,000.
  2. $1,000 × 0.75 = $750.

Answer: $750

Streaks multiply

The chance of several independent losses in a row.

  1. Write the loss chance as a decimal: 40% is 0.4.
  2. Multiply it by itself once per loss in the streak.
  3. Turn the result back into a percent.
worked example

Example: You lose 50% of bids, independently of each other. What is the chance of losing your next 3 bids in a row? Give a percent rounded to 1 decimal place.

  1. 0.5 × 0.5 × 0.5 = 0.125.
  2. That is 12.5%.

Answer: 12.5%

Spread grows with √n

Many independent, identical bets: the swing of the total, or of the average.

  1. Independent bets add their variances, the squares of their spreads.
  2. So the total's standard deviation is √n times one bet's.
  3. The average divides the total by n, so its spread is 1/√n of one bet's.
worked example

Example: One bet wins or loses $200 with equal chance. You make 25 independent bets like it. What is the standard deviation of your total result, in dollars?

  1. Variances add: 25 × 200², so take the square root.
  2. √25 × $200 = 5 × $200 = $1,000.

Answer: $1,000

Tips by skill

  • TipUp then down: Turn each percent into a multiplier, such as × 1.2 or × 0.8, and apply them in order. Never add the percents.
  • TipMean vs median: One extreme value changes the total, so the mean jumps. The middle of the sorted list stays put, so the median barely moves.
  • TipLosing streaks: Chances that must all happen multiply. Raise the loss chance, as a decimal, to the length of the streak.
  • TipSpreading bets: For the total, the spread grows by √n. For the average per bet, it shrinks by √n. Never by n itself.

Watch out for

  • Assuming +50% and −50% cancel. The loss hits the bigger amount, so $100 ends at $75.
  • Adding the chances in a streak: 40% four times is not 160%. Chances that must all happen multiply.
  • Multiplying the spread by n instead of √n. Spreads add as squares, so 16 bets swing 4 times as much, not 16.
  • Expecting the median to jump with one outlier. It only looks at the middle of the sorted list.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

No rounds yet.