Risk & the long run
Volatility, losing streaks, spreading bets, and outliers.
In your head, jot if needed · no calculator why?
Opens at level 35. You're level 1. You can read and practice here now.
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
- Turn each percent into a multiplier: +25% is × 1.25, −25% is × 0.75.
- Apply them in order.
- Equal ups and downs always end below the start.
worked example
$800 gains 25%, then loses 25%. What is it worth now?
- $800 × 1.25 = $1,000.
- $1,000 × 0.75 = $750.
Answer: $750
Streaks multiply
- Write the loss chance as a decimal: 40% is 0.4.
- Multiply it by itself once per loss in the streak.
- Turn the result back into a percent.
worked 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.
- 0.5 × 0.5 × 0.5 = 0.125.
- That is 12.5%.
Answer: 12.5%
Spread grows with √n
- Independent bets add their variances, the squares of their spreads.
- So the total's standard deviation is √n times one bet's.
- The average divides the total by n, so its spread is 1/√n of one bet's.
worked 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?
- Variances add: 25 × 200², so take the square root.
- √25 × $200 = 5 × $200 = $1,000.
Answer: $1,000
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.
practice
Up then down
worked example
An investment of $9,500 loses 10%, then gains 10%. What is it worth now?
Answer: $9,405.00
- $9,500 × 0.9 = $8,550, then $8,550 × 1.1 = $9,405.
- The 10% gain works on the smaller amount, so equal ups and downs lose money.
Mean vs median
worked example
A shop's 9 orders today were each $30. Then one order for $15,000 comes in. Which stays the same: the mean or the median?
- the mean
- both equally
- neither
- the median
Answer: the median
- The mean jumps from $30 to $1,527; the median stays $30, because the middle value is still $30.
- When a few values are extreme, the median describes the typical one better.
Losing streaks
worked example
30% of your sales pitches fail, independently of each other. What is the chance that your next 4 pitches all fail? Give a percent rounded to 1 decimal place.
Answer: 0.8%
- Independent chances multiply: 0.3⁴ = 0.0081, which is 0.81%, about 0.8%.
Spreading bets
worked example
One bet wins or loses $120 with equal chance. You make 9 independent bets like it. How does the standard deviation of your total result compare with one bet's?
- 3 times as big
- 1/3 as big
- 9 times as big
- the same
Answer: 3 times as big
- Variances of independent bets add: 9 × 120². The standard deviation is the square root: √9 × $120 = $360, which is 3 times one bet's.