Scratch-Off (Untimed, 7 Questions)

How it works

You start with a $100 bankroll and pick a ticket priced at $1, $5, $10, $20, $50, or $100. The ticket flips over to reveal its true prize table before you commit to anything - three outcomes (a $0 tier, a mid tier, and a jackpot tier) with exact fractional odds out of 8 or 16. Real lotteries hide this table; the game shows it to you because reading it is the whole skill.

Before you can scratch, you answer a set of probability questions about the ticket you are holding: expected value, Bayes / conditional probability, no-replacement batch odds, combinatorics, the geometric distribution (expected tickets to first win), and an occasional classic puzzle (Monty Hall, a birthday-problem variant, coupon collector). Then comes the decision stage: buy anywhere from 0 to 10 tickets of this design at this price. Choosing 0 walks away and refunds the ticket price in full - the game still shows what the ticket would have paid, for the record.

This variant runs 7 questions per ticket with no timer at all - work each answer until you are sure. The session ends only when you cash out (keeping your bankroll) or it hits $0, which the game names for what it is: gambler's ruin.

How scoring works

A ticket that shows multiplier m pays exactly m times the ticket price - the three tiers are $0, a mid prize of 1x or 2x, and a jackpot of 4x, 5x, 6x, or 8x on a normal ticket, or 8x, 10x, or 12x on a rare mispriced one. The $0 tier always covers between 50% and 80% of the probability mass.

70% of generated tickets are fair-ish: expected value at or below 1x the price, a genuine house edge. 30% are hot: EV strictly above 1x, mispriced in the player's favor. The split is constant and is never adjusted based on whether you are winning or losing.

Walking away at the quantity step (choosing 0) refunds your ticket price - a guaranteed EV of exactly 1.0x, better than most tickets in the game.

Compute EV before the buy decision, every time

The flip reveal hands you the full distribution, so EV is one line of arithmetic: multiply each prize by its probability and add. On a $10 ticket showing P($0) = 5/8, P(2x = $20) = 2/8, P(4x = $40) = 1/8: EV = (2/8)($20) + (1/8)($40) = $5.00 + $5.00 = $10.00 - exactly 1.0x, the best a fair-side ticket ever gets. A harsher fair shape like P($0) = 6/8, P(1x = $10) = 1/8, P(5x = $50) = 1/8 gives EV = (1/8)($10) + (1/8)($50) = $7.50, a 25% house edge.

The end-of-session screen states the game's own summary: most tickets pay out about 7/8 of the price - a 12.5% house edge. Any ticket whose EV multiple comes out at or below 1 is a losing or at best break-even bet, and the refund option (buy 0) strictly dominates buying it.

Size up only on hot tickets

About 3 tickets in 10 are generated with EV above 1x - jackpot multipliers of 8x, 10x, or 12x with a zero tier capped near 60%. Those are the only +EV bets in the game. When the arithmetic says EV > price, the max-quantity rule (up to 10 tickets, limited by bankroll) is your edge: each additional ticket adds the same positive expected profit. When EV is at or below the price, the right quantity is 0 - take the refund.

Note what buying more does NOT do: it never changes the per-ticket edge. Ten tickets of a 0.75x design have EV 7.5x the price, not a better shot at the jackpot per dollar. Volume scales the edge you already have, in whichever direction it points.

Use the missing clock to show full work

With no timer, treat every question like a whiteboard round: write the distribution, name the event, produce the exact fraction. The answer box accepts fractions like 3/8 or decimals like 0.375, and tolerances are tight (typically 0.01 on probabilities), so exact fractions are safer than rounded decimals.

The seven-question set always covers all six topics once (EV, Bayes, batch conditional, combinatorics, geometric, classic) plus one repeat, with 1 to 3 questions posed in formal notation. Untimed mode is where to build the habit of translating P(B|A) = P(B)/P(A) and E[X] = 1/p into arithmetic without pressure, before graduating to the timed variants.

A worked example

You buy a $10 ticket. The flip reveal shows: $0 with probability 5/8, $10 (1x) with probability 2/8, $40 (4x) with probability 1/8.

EV = (5/8)(0) + (2/8)($10) + (1/8)($40) = $2.50 + $5.00 = $7.50. Expected profit = $7.50 - $10.00 = -$2.50, a 25% house edge. This is a fair-side shape - decline it.

Sample questions on this ticket: P(win nothing) = 5/8. P(top prize | win something) = (1/8) / (3/8) = 1/3. P(win something) = 3/8, so expected tickets to first win = 8/3, about 2.67.

At the decision stage you set quantity to 0. Your $10 is refunded and your bankroll is unchanged - an EV of exactly $10.00 versus the $7.50 of playing. Walking away outperformed the ticket by $2.50 of expectation, which is the lab's title acted out.

Common mistakes

• Buying before doing the EV multiplication. The table is face-up - a ticket that pays 7/8 of its price loses you 12.5 cents per dollar in expectation, every time.

• Answering W/N on the batch question. After a non-winner is removed, W winners remain among N-1 tickets: the answer is W/(N-1).

• Sizing up on a fair ticket to chase the jackpot. Quantity multiplies expected loss just as faithfully as expected gain.

• Ignoring the walk-away option. Buying 0 refunds the full price - a guaranteed 1.0x that beats roughly 70% of tickets generated.

• Rounding early. Tolerances are around 0.01 on probability answers - enter 5/8 rather than 0.6.

Why interviews test this

Expected value over an explicit discrete distribution is the single most common opener in quant interviews - it tests whether you instinctively price a bet instead of reacting to its best case. The follow-ups here mirror real follow-ups: conditional probability without replacement (does the candidate update on removed outcomes?), the geometric distribution (time to first success), and Bayes on nested events.

The deeper lesson interviewers probe is bet selection and sizing: a trader who can compute EV but still takes -EV bets, or who sizes the same regardless of edge, fails the job even if the arithmetic is right. Declining the bet - quantity 0 - is frequently the correct answer, and saying so out loud is a strong interview signal.