Susquehanna's (SIG) Quant phone and onsite loops are famous for probability density — a single 30-minute call can pack 8-10 probability problems, all answered verbally, no calculator, no scratch paper (whiteboard maybe in onsite). This bank groups the 18 most frequent SIG probability questions into 7 thinking models, with a 30-second template and a Python simulation per category. Companion piece to the SIG OA three-track guide — this is the pocket bank for phone screen and onsite season.
SIG loop snapshot
Phone Round 1 HR + projects + 2-3 warm-up probability (30 min)
Phone Round 2 Specialist phone: 6-10 probability bombs (30 min)
Onsite Round 1 6 probability + expectation, mental-math timed (45 min)
Onsite Round 2 Trading game / market making sim (45 min)
Onsite Round 3 Behavioral + culture fit (30 min)
Phone Round 2 is the single hardest filter — miss two and you're cut. The questions almost always live inside the 7 categories below.
Category 1 — Conditional probability (highest frequency)
Q1.1 Two-headed coin
A bag holds 99 fair coins + 1 two-headed coin. You pick one and flip 7 heads in a row. Conditional probability it's the two-headed coin?
Bayes:
- P(D) = 1/100, P(7H | D) = 1
- P(F) = 99/100, P(7H | F) = 1/128
- P(D | 7H) = (1/100·1) / (1/100·1 + 99/100·1/128) = 128 / (128 + 99) ≈ 56.4%
def bayes_two_headed():
p_d = 1/100 * 1
p_f = 99/100 * (1/2)**7
return p_d / (p_d + p_f)
Q1.2 International trips
Last year you took two trips; the December one was international. What's the conditional probability both were international?
Spec is ambiguous — does "December was international" come from observation or sampling? SIG wants you to clarify proactively. If "December was international" is given, then P(other is international) = P(international), independent of December.
Category 2 — Penney's Game / sequence probability
Q2.1 HHT vs HTH
A picks HHT, B picks HTH. First sequence to appear wins.
Memorize: A wins with probability 2/3. Penney's Game's classic counterintuitive result. Markov chain:
state 0 -> H -> state 1
state 1 -> H -> state 2 (HH)
state 1 -> T -> state 0
state 2 -> H -> state 2
state 2 -> T -> A wins (HHT)
B trying to wait for HTH frequently bumps into HHT first.
import random
def simulate_penney(n=200000):
a_wins = 0
for _ in range(n):
seq = []
while True:
seq.append(random.randint(0,1)) # 1=H, 0=T
s = "".join("H" if x else "T" for x in seq[-3:])
if s == "HHT":
a_wins += 1; break
if s == "HTH":
break
return a_wins / n # ≈ 0.667
Category 3 — Waiting time paradox / parallel waits
Q3.1 Three bus routes
Bus A arrives at U(0,10), B at U(0,20). Average wait?
Independent arrivals, wait = min(T_A, T_B). For two routes, E[min(U(0,a), U(0,b))] (a≤b) = a/2 - a²/(6b).
Different frequencies + random passenger arrival = inspection paradox. SIG wants you to distinguish "passenger-perceived gap" from "true gap."
Category 4 — Geometric probability
Q4.1 Three points on a circle, obtuse triangle
Three random points on a circle form a triangle. Probability it's obtuse?
Memorize: 3/4. Intuition: P(acute) = 1/4 because all three must avoid the same half-circle on either side; P(obtuse) = 3/4.
Q4.2 Max regions of a circle cut by n lines
n straight lines, max regions in a circle?
Formula: R(n) = 1 + n + C(n,2) = (n² + n + 2) / 2.
Q4.3 Cube paint
Paint the outside of a cube blue, then cut it 2 ways into 27 small cubes. Probability that a random small cube has a yellow (original) top face?
For each small cube, "top face is yellow" depends on its layer. Bottom layer (9) and middle layer (9) cubes have unpainted top faces; top layer (9) has painted tops. The exact answer hinges on the cut definition — SIG loves these geometric edge details.
Category 5 — Expectation (with branches)
Q5.1 Replace until all blue
Bag: 2 red, 1 blue. After each draw, replace with a blue ball. Expected draws until all blue?
States by red count: 2 → 1 → 0.
- 2 → 1: P(draw red) = 2/3, geometric mean = 3/2
- 1 → 0: P(draw red) = 1/3, mean = 3
- Total = 1.5 + 3 = 4.5
Q5.2 Alternating dice game
A and B roll dice alternately (A first). Game ends when A rolls a 6. Expected total rolls?
State machine + geometric distribution. Each "round" = A roll + B roll = 2 rolls; A succeeds at 1/6.
E = 6 × 2 - 1 = 11 (the final roll is A's 6; B doesn't roll after).
Category 6 — Game theory / decision thresholds
Q6.1 Kelly-style bet
Stay → $1000. Answer correctly → $4000. Answer incorrectly → only $250. Minimum p to make answering worthwhile?
E = 4000p + 250(1-p) ≥ 1000 → 3750p ≥ 750 → p ≥ 20%.
Q6.2 Minimum win rate to call
You and friend each blind $10. Friend raises to $20. If you fold, you lose your blind. Minimum p to call?
- Fold: lose 10
- Call: win 30 with prob p, lose 30 with prob 1-p
- E(call) ≥ E(fold) → 30p - 30(1-p) ≥ -10 → p ≥ 33.3%
Category 7 — Round table / arrangement probability
Q7.1 Circular adjacency
8 people seat around a round table; 3 selected. Probability that at least 2 of those 3 are adjacent?
Complement: P(all three pairwise non-adjacent).
- Total ways C(8,3) = 56
- Non-adjacent ways on a circle = 8/(8-3) × C(8-3,3) = 8/5 × 10 = 16
- P(non-adjacent) = 16/56 = 2/7
- P(at least 2 adjacent) = 5/7 ≈ 71.4%
Mental-math discipline (the 30-second rule)
SIG phone interviews give 30-60 seconds per question. Overrunning is almost certain failure. Three rules:
- Answer first, derive second — under 30 seconds, lead with the number + a one-line intuition; if the interviewer accepts, move on
- Memorize 8 standard results — Penney HHT 2/3, obtuse-on-circle 3/4, two-headed coin 56.4%, Kelly 20%, cube paint top-face fraction, etc.
- Clarify on ambiguous prompts — burn 5 seconds asking, save 30 seconds of misdirection
OA → Phone → Onsite cadence
| Stage | Type | Prep focus |
|---|---|---|
| OA | CodeSignal coding + math | LeetCode Medium + probability speed |
| Phone R1 | Resume + 2-3 warm-ups | 90-second self-intro |
| Phone R2 | 6-10 probability bombs | The 18 questions above, fluently |
| Onsite R1 | 6 + mental math | 30-second rule drilling |
| Onsite R2 | Trading game | Market-making intuition |
OA assist plug-in points for SIG
SIG sits in the "memorizable bank" lane of OA assist (OA assist (OA live support)) — the core 18-25 problems cover ~80% of the prompts. Standard cadence:
- Bank identification — invitation screenshot tells us which of the OA three tracks (coding / math / trading game) you're on
- Timed mock — 1 hour mental-math drills + 1 hour OA coding daily
- Live cueing — phone day, backstage Bayes / Penney shortcuts piped through
- Debrief — log every miss after each round, re-walk the template
- Onsite trading-game drill — 2-3 hours of market-making sim
FAQ
Q1: Really no calculator on SIG probability? A: 100% verbal on phone. Onsite some rounds allow whiteboard but mental math still dominates.
Q2: My math background is average — can I prep through this? A: Yes. SIG has high overlap; memorize 18-25 standard results + templates and you'll hit ~70%+.
Q3: NG salary at SIG? A: Quant Trader / QR base ~$200K, sign-on $100K-$200K, year-1 bonus 50-100% of base depending on the loop score.
Q4: Can I reapply after rejection? A: Yes, typically a 1-year cool-down. Drill the bank before reapplying.
Q5: When's the latest stage to start OA assist? A: Day-of OA invitation is best. Pre-Phone-R1 also works. Phone R2 must have the bank locked.
Closing
SIG looks like a probability test, but really it tests "30-second mental math + template reuse" as a special skill. If you're prepping SIG / Akuna / Optiver / Jane Street, message WeChat Coding0201 with your current phone / onsite stage. We'll identify the bank first, then schedule OA assist accordingly.
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Contact
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