Chapter 1 — Problem Set

The Sample Space

Problems drawn from Feller (1968) Ch. 1, with quant finance extensions sourced from Zhou (2008) and Joshi (2008).

Fundamentals

1.1 Among the digits 1, 2, 3, 4, 5 first one is chosen, then a second from the remaining four. All twenty outcomes equally likely. Find the probability that an odd digit is chosen (a) first; (b) second; (c) both times.

1.2 A coin is tossed until the same result appears twice in succession. Assign probability 1/2^n to each outcome requiring n tosses. (a) Describe the sample space. (b) \mathbb{P}(\text{ends before toss 6}). (c) \mathbb{P}(\text{even number of tosses}).

1.3 Two dice are thrown. A = sum is odd, B = at least one ace. Find \mathbb{P}(AB), \mathbb{P}(A \cup B), \mathbb{P}(A\bar{B}).

1.4 A point is chosen at random inside a circle of radius R. Find the probability it is (a) within distance r < R from the centre; (b) closer to the centre than to the boundary.

1.5 (Polya urn) An urn has b black and r red balls. Draw one, replace it plus c balls of the same colour. Show that \mathbb{P}(\text{second draw red}) = \mathbb{P}(\text{first draw red}), but the draws are not independent.

1.6 (Derangements) n letters are randomly placed in n envelopes. Use inclusion-exclusion to find \mathbb{P}(\text{at least one correct}) and show it approaches 1 - 1/e as n \to \infty.

Quant Finance

QF-1 (Gambler’s Ruin) A trader starts with $3 and makes fair $1 bets, stopping at $0 or $6. (a) Find her ruin probability. (b) Redo for win probability p = 0.45. (c) What edge (positive p - 0.5) is needed to bring ruin probability below 10%?

QF-2 (CRR Binomial) S_0 = 100, u = 1.1, d = 0.9, 3 periods, r = 0. (a) List all 8 paths. (b) Find \mathbb{P}(S_3 > 105) under real-world p = 0.6. (c) Find risk-neutral \tilde{p} and recompute.

QF-3 (Bayesian Regime) Prior \mathbb{P}(\text{trending}) = 0.4. Signal fires on 70% of trending days, 20% of mean-reverting days. (a) Posterior after one signal. (b) After two signals. (c) Prior needed for 50% posterior after one signal.

QF-4 (Diversification limits) n assets each default independently with p = 0.05. (a) \mathbb{P}(\text{at least one default}) as n \to \infty. (b) Redo with perfect correlation. (c) What does this say about CDO tranching?

Challenge

C-1 (Simpson’s Paradox) Construct a numerical example where treatment A beats treatment B in both subgroups but loses overall. Express in terms of conditional vs. marginal probabilities.

C-2 (Monty Hall — general) n doors, host reveals k empty doors. Probability of winning by switching?

References

Feller, William. 1968. An Introduction to Probability Theory and Its Applications. 3rd ed. Vol. 1. Wiley.

Joshi, Mark S. 2008. Quant Job Interview Questions and Answers. Pilot Whale Press.

Zhou, Xinfeng. 2008. A Practical Guide to Quantitative Finance Interviews. Lulu Press.

--- title: "Chapter 1 — Problem Set" subtitle: "The Sample Space" toc: true bibliography: ../../references.bib --- Problems drawn from @feller1968 Ch. 1, with quant finance extensions sourced from @zhou2008 and @joshi2008. --- ## Fundamentals **1.1** Among the digits $1, 2, 3, 4, 5$ first one is chosen, then a second from the remaining four. All twenty outcomes equally likely. Find the probability that an odd digit is chosen (a) first; (b) second; (c) both times. **1.2** A coin is tossed until the same result appears twice in succession. Assign probability $1/2^n$ to each outcome requiring $n$ tosses. (a) Describe the sample space. (b) $\mathbb{P}(\text{ends before toss 6})$. (c) $\mathbb{P}(\text{even number of tosses})$. **1.3** Two dice are thrown. $A$ = sum is odd, $B$ = at least one ace. Find $\mathbb{P}(AB)$, $\mathbb{P}(A \cup B)$, $\mathbb{P}(A\bar{B})$. **1.4** A point is chosen at random inside a circle of radius $R$. Find the probability it is (a) within distance $r < R$ from the centre; (b) closer to the centre than to the boundary. **1.5** *(Polya urn)* An urn has $b$ black and $r$ red balls. Draw one, replace it plus $c$ balls of the same colour. Show that $\mathbb{P}(\text{second draw red}) = \mathbb{P}(\text{first draw red})$, but the draws are not independent. **1.6** *(Derangements)* $n$ letters are randomly placed in $n$ envelopes. Use inclusion-exclusion to find $\mathbb{P}(\text{at least one correct})$ and show it approaches $1 - 1/e$ as $n \to \infty$. --- ## Quant Finance **QF-1** *(Gambler's Ruin)* A trader starts with \$3 and makes fair \$1 bets, stopping at \$0 or \$6. (a) Find her ruin probability. (b) Redo for win probability $p = 0.45$. (c) What edge (positive $p - 0.5$) is needed to bring ruin probability below 10%? **QF-2** *(CRR Binomial)* $S_0 = 100$, $u = 1.1$, $d = 0.9$, 3 periods, $r = 0$. (a) List all 8 paths. (b) Find $\mathbb{P}(S_3 > 105)$ under real-world $p = 0.6$. (c) Find risk-neutral $\tilde{p}$ and recompute. **QF-3** *(Bayesian Regime)* Prior $\mathbb{P}(\text{trending}) = 0.4$. Signal fires on 70% of trending days, 20% of mean-reverting days. (a) Posterior after one signal. (b) After two signals. (c) Prior needed for 50% posterior after one signal. **QF-4** *(Diversification limits)* $n$ assets each default independently with $p = 0.05$. (a) $\mathbb{P}(\text{at least one default})$ as $n \to \infty$. (b) Redo with perfect correlation. (c) What does this say about CDO tranching? --- ## Challenge **C-1** *(Simpson's Paradox)* Construct a numerical example where treatment A beats treatment B in both subgroups but loses overall. Express in terms of conditional vs. marginal probabilities. **C-2** *(Monty Hall — general)* $n$ doors, host reveals $k$ empty doors. Probability of winning by switching?