Two fair dice are rolled. What is the probability that they sum to 7?
Six of the 36 equally likely outcomes sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So .
Daily five
Five questions picked fresh each day from across the bank, from warm-up to hard. Solve all five for a +25 bps bonus.
0 of 5 solved
On average, how many rolls of a fair die does it take to roll the first 6?
The number of trials until the first success is geometric with , so its mean is .
What is the expected number of fair coin flips until you see two heads in a row?
Let be the expected flips from scratch and after one head. and . Solving gives .
Three cards are drawn from a standard deck without replacement. What is the probability all three are hearts? (decimal)
.
What is the smallest number of people for which the chance that two share a birthday exceeds 50%? (365 equally likely days)
The probability all birthdays differ is . It first drops below 0.5 at (about 0.493).
A stick is broken at two points chosen uniformly at random. What is the probability the three pieces can form a triangle?
A triangle is possible exactly when no piece is longer than half the stick. In the unit square of break points that region has area .
X and Y are independent Uniform(0, 1). What is the expected value of max(X, Y)?
, so the density is and . In general the max of uniforms has mean .
You roll a die and are paid its face value in dollars. You may reroll once, but must then accept the second roll. With optimal play, what is the game worth?
A reroll is worth 3.5, so keep 4, 5 or 6 and reroll 1–3. Value .
On average, how many rolls of a fair die does it take to see every face at least once?
Coupon collector: after faces are seen, the wait for a new one is geometric with mean . Total .
What is the probability of at least one six in four rolls of a fair die? (decimal)
Use the complement: . (The Chevalier de Méré problem.)
Monty Hall: you pick one of three doors, the host (who knows where the car is) opens another door showing a goat and offers a switch. What is your win probability if you switch?
Your first pick is right with probability 1/3, and the host’s action never changes that. Switching wins whenever your first pick was wrong: 2/3.
A family has two children and at least one is a boy. What is the probability both are boys?
Of the equally likely BB, BG, GB, GG, the condition removes GG. One of the remaining three is BB: 1/3.
A drawer holds 10 black and 10 white socks. In the dark, how many must you take to guarantee a matching pair?
Pigeonhole principle: with two colours, three socks must include two of the same colour.
You start with 2 dollars and bet 1 on fair coin flips until you reach 5 or go broke. What is the probability you reach 5?
For a fair game your wealth is a martingale, so .
Ten people throw their hats in a pile and each takes one at random. What is the expected number who get their own hat?
By linearity of expectation, each person gets their own hat with probability 1/10, and ten such indicators sum to an expected 1 — for any number of people.
X and Y are independent standard normals. What is P(X > 0 and X + Y > 0)?
By rotational symmetry the answer is the angle between the half-planes and divided by . That angle is , giving .
One of 100 coins is double-headed. You pick one at random, flip it 10 times and see 10 heads. What is the probability you hold the double-headed coin? (decimal)
Posterior .
A crash signal fires on 95% of real crashes and 10% of normal weeks. Crashes occur in 2% of weeks. Given that it fired, what is the probability of a crash? (decimal)
. Rare events make even accurate signals mostly false alarms.
You have 100 independent daily returns with a standard deviation of 2%. What is the standard error of the mean daily return, in percent?
Corr(X, Y) = 0.6 and Y’s volatility is twice X’s. What is the OLS slope of Y regressed on X?
A strategy has an annualised Sharpe ratio of 1.0 over 4 years of data. Approximately what is its t-statistic?
An equity strategy has 1.2% daily volatility. What is its annualised volatility in percent (252 trading days)?
You combine 9 uncorrelated strategies, each with Sharpe 0.5, at equal risk. What is the Sharpe ratio of the combination?
Means add linearly while volatilities add in quadrature, so the Sharpe scales by : .
You test 20 strategies with no real edge at the 5% significance level. How many do you expect to look significant?
. Selecting it would be a pure false positive.
Same 20 worthless strategies at 5% significance, tested independently. What is the probability at least one looks significant? (decimal)
. Search widely enough and a "discovery" is almost guaranteed.
Two assets each have 10% volatility and correlation 0.5. What is the volatility of a 50/50 portfolio, in percent?
(in units of ), so .
You regress the price of one coin on the price of an unrelated coin and get R² = 0.85. What is the most likely explanation?
Regressing one random walk on another routinely produces high with no real relationship. Regress returns, or test residuals for stationarity (cointegration).
A spread follows an AR(1) with daily coefficient φ = 0.95. What is its half-life in days?
A setup wins 40% of the time for +3R and loses 60% of the time for −1R. What is its expectancy per trade, in R?
. A low win rate can still be very profitable.
A strategy risks 1 to make 2 on every trade. What win rate, in percent, is break-even (ignoring costs)?
After a 25% loss, what percentage gain is needed to get back to even?
. Losses and recoveries are asymmetric.
An even-money bet wins 60% of the time. What fraction of wealth, in percent, does Kelly say to bet?
A strategy has expected excess return 8% and volatility 20% a year. What is the full-Kelly leverage?
. Most practitioners would run half of that or less.
What long-run growth rate do you get betting exactly twice the Kelly fraction?
Growth is a parabola that crosses zero at , exactly twice Kelly.
An asset has an arithmetic mean return of 10% and volatility of 30% a year. Approximately what is its compound growth rate, in percent?
A perp pays 0.01% funding every 8 hours. What does that annualise to, in percent (365 days, no compounding)?
Equity is 50,000. You risk 1% per trade, entering at 100 with a stop at 95. How many units do you buy?
Risk is 500 and the stop is 5 away, so units.
A 1,000,000 position has normally distributed daily returns with 2% volatility and zero mean. What is its one-day 99% VaR, in thousands?
A one-day VaR is 10,000. Using the square-root-of-time rule, what is the 10-day VaR?
. The rule assumes independent, identically distributed returns.
Using the rule of 72, roughly how many years does money take to double at 9% a year?
years. (Exactly, .)
Risking 1% of current equity per trade, what is the total drawdown, in percent, after 10 straight losses?
. Risking a fixed fraction of current equity shrinks bets after losses.
Estimate the price of a one-year at-the-money call on a 100 stock with 20% volatility and zero rates.
. The exact Black–Scholes value is about 7.97.
A call struck at 100 trades at 10. The stock is at 100 and rates are zero. What should the put with the same strike and expiry cost?
Put–call parity: .
A 95-strike call costs 8 with the stock at 100 and zero rates. What is the 95-strike put worth?
Where is an option’s gamma largest?
Gamma peaks at the money and grows without bound as .
A long straddle (long call + long put, same strike) is:
Both legs are long options, so the position is long gamma and vega and pays theta. It profits from a big move or a rise in implied volatility.
Approximately what is the delta of a short-dated at-the-money call?
and at the money for short maturities, so .
An ATM option with 4 months to expiry and one with 1 month have the same implied vol. Roughly how many times more does the 4-month option cost?
ATM value scales with , and .
An index has 20% implied volatility. What one-day move, in percent, does that imply (252 trading days)?
. A quick "rule of 16" divides annual vol by 16.
You buy an option at 25% implied vol and delta-hedge it continuously. Realised volatility turns out to be 20%. Ignoring costs, you:
A delta-hedged long option earns . Realised below implied means the theta you pay exceeds what gamma earns.
How many times do the hour and minute hands of a clock overlap in 12 hours?
The minute hand laps the hour hand once every hours, so 11 times in 12 hours (12:00 counts once).
100 closed lockers. Student k toggles every k-th locker, for k = 1 to 100. How many lockers end up open?
A locker is toggled once per divisor. Only perfect squares have an odd number of divisors, and there are 10 squares up to 100.
A bat and ball cost 1.10 together. The bat costs 1.00 more than the ball. What does the ball cost?
. The intuitive 0.10 is wrong.
How many trailing zeros does 100! have?
Count factors of 5: .
Two trains 100 km apart head towards each other at 50 km/h each. A fly zips between them at 75 km/h until they meet. How far does the fly travel?
The trains meet after one hour, so the fly flies for one hour: 75 km. No infinite series needed.
25 horses, a track that races 5 at a time, no timer. What is the minimum number of races to find the 3 fastest?
Five heats, then a race of the five winners (6). Only 5 horses remain candidates for 2nd and 3rd; race them (7).
100 ants are placed randomly on a 1 m stick, each walking at 1 m/min in a random direction. Ants that meet both reverse. What is the longest time before every ant has fallen off?
Two ants bouncing is indistinguishable from two ants passing through each other. Each "ant" then walks straight off in at most 1 minute.
Among 12 coins one is counterfeit — heavier or lighter, you don’t know which. With a balance scale, what is the minimum number of weighings to find it and tell whether it is heavy or light?
There are 24 possibilities and each weighing has 3 outcomes; , and a three-weighing scheme exists.
Eight balls, one slightly heavier. What is the minimum number of balance-scale weighings to find it?
Weigh 3 vs 3. If they balance, weigh the remaining 2; if not, weigh 1 vs 1 from the heavy side. Two weighings: .
What is 1 + 2 + 3 + … + 100?
.
17 × 23 = ?
.
48 × 52 = ?
.
What is 1/7 as a percentage, to two decimal places?
.
What is 15% of 640?
10% is 64 and 5% is 32, so 96.
0.35 × 0.6 = ?
, with three decimal places: 0.210.
Estimate √2000 to two decimal places.
, so .
99² = ?
.
3/8 + 5/12 = ? (decimal)
Common denominator 24: .
125 × 64 = ?
, and , so 8000.
Estimate e^0.1 to three decimal places.
.
A price rises 10% and then falls 10%. What is the total change, in percent?
, a 1% loss.
Make a market on the sum of two fair dice. What is the fair value you should centre your quote on?
Each die has mean 3.5, so the sum has mean 7. Quote a spread around it, e.g. 6.5 at 7.5.
A game pays the square of a fair die roll. What is its fair value?
. Note it exceeds — the gap is the variance.
You make a market on the number of heads in 10 fair coin flips. What is the standard deviation of the outcome?
Binomial: . It tells you how wide your quotes should be.
You quote 48 bid, 52 offer. A counterparty buys your offer again and again. What should you do?
Persistent one-sided flow is information (adverse selection). Move your quotes up and possibly widen until the flow balances.
In the Avellaneda–Stoikov model you are long a large inventory. Your reservation price is:
. With the reservation price sits below the mid, so you quote lower to shed inventory.
A stock is quoted 99.98 bid, 100.02 offer. What does a round trip of 10,000 shares at market cost in spread alone?
You buy at 100.02 and sell at 99.98: .
Best bid 100.00 for 900, best ask 100.02 for 100. What is the micro-price?
. Heavy bid size pulls fair value towards the ask.
Using the square-root law with Y = 1, what is the expected impact of buying 1% of daily volume in an asset with 4% daily volatility, in percent?
Today’s five: done.
Come back tomorrow for a new set. In the meantime, try the mental math sprint or work through the full bank.