Probability & EV Practice
Dice, coin and card probability plus expected-value calculation. Today's free drill comes from a 180-day pool. No signup. Runs in your browser.
15 questions · 3 minutes · today's set
Dice, coin, and card probability plus quick expected-value calculations. +1 per correct answer, −1 per wrong answer, skips are free. Everyone gets the same 15 questions today.
0.25 or 1/4).Six probability question types
These six families introduce the calculations used in our daily drill. Each example assumes fair dice or coins and a standard deck unless stated otherwise:
| Dice-sum probability | e.g. P(sum of two dice = 7): 6 of the 36 equally likely outcomes work, so 6/36 = 1/6. |
|---|---|
| Die-roll expected value | e.g. EV of one fair 6-sided roll: (1+2+3+4+5+6)/6 = 3.5. |
| Coin-flip runs | e.g. P(3 heads in a row): each flip is independent, so (1/2)³ = 1/8. |
| Card draws | e.g. P(drawing an Ace from a standard 52-card deck): 4/52 = 1/13. |
| Binary-bet EV | e.g. a bet paying +$10 with probability 0.4 and −$5 with probability 0.6 has EV = 0.4(10) + 0.6(−5) = +$1 per play, before any additional costs. |
| "At least one" / "never" probability | e.g. P(at least one 6 in 4 rolls) is easiest via the complement: 1 − (5/6)⁴ ≈ 0.52. |
Worked explanations and common errors
1. Two dice: define the sample space
Question: Two independent fair six-sided dice are rolled. What is the probability that their sum is 7? Answer: 1/6. There are 6 × 6 = 36 equally likely ordered outcomes. Six work: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Divide 6 by 36.
Common error: treating the 11 possible sums as equally likely. Sum 2 has only (1,1), while sum 7 has six outcomes. If the first die is known to be 2, the conditional calculation instead uses the six possible second-die results; only 5 completes the sum.
2. Die-roll EV: weight each payout
Question: A fair six-sided die pays $2 per pip. What is the expected payout? Answer: $7. The six equally likely payouts are $2, $4, $6, $8, $10 and $12. Their average is $42/6 = $7, also 2 × 3.5. If entry costs $3, expected net profit is $7 − $3 = $4.
Common error: confusing payout with profit, or expecting a single roll to equal the expectation. This game never pays exactly $7 on one roll; EV describes the probability-weighted average.
3. Coin flips: specify the number of trials
Question: You flip a fair coin exactly three times, independently. What is the probability of three heads? Answer: 1/8. Each flip has probability 1/2 of heads, so multiply: (1/2) × (1/2) × (1/2) = 1/8. HHH is one of eight equally likely length-three outcomes.
Common error: reading this as three consecutive heads somewhere in a longer series. That is a different event. For a coin with heads probability p, the exactly-three-flips calculation becomes p³, assuming independence.
4. Card draws: replacement changes the denominator
Question: What is the probability of an Ace on one uniform draw from a standard 52-card deck? Answer: 1/13. Four cards qualify, so 4/52 = 1/13. For two Aces without replacement, multiply 4/52 by 3/51 to get 1/221. With replacement and a fresh uniform draw, it is (4/52)² = 1/169.
Common error: using the same numerator and denominator for the second draw after removing an Ace. State whether cards are replaced before calculating.
5. Binary-bet EV: keep losses negative
Question: A game changes your balance by +$10 with probability 0.4 and −$5 with probability 0.6. What is its EV? Answer: +$1 per play. Multiply each net outcome by its probability: 0.4 × 10 + 0.6 × (−5) = 4 − 3 = 1. If winning probability is p, EV is 10p − 5(1 − p) = 15p − 5, which is zero at p = 1/3.
Common error: adding the loss as a positive amount or subtracting an entry stake twice when the given outcomes are already net. Positive EV does not guarantee a profit on a particular play or account for risk constraints.
6. At least one success: use the complement
Question: What is the probability of at least one 6 in four independent rolls of a fair die? Answer: 671/1296, about 0.5177. The probability of no 6 is (5/6)⁴ = 625/1296. Subtract from 1: (1296 − 625)/1296 = 671/1296. This includes one, two, three or four sixes.
Common error: calculating 4 × (1/6), which double-counts outcomes with more than one 6. Exactly one 6 is a different event with probability 4 × (1/6) × (5/6)³ = 125/324.
Next: try the timed 15-question probability drill. To practice choosing an action from the sign of EV, use the Robot Games Play/Pass drill.
Probability and EV in later interview rounds
Interviewers use the same probability and EV tools in brain-teaser rounds, and EV explains why negative marking changes test strategy. The full prep guide covers both in more depth. The Robot Games drill applies the same EV-sign judgment to a timed Play/Pass decision instead of a typed number.
Number sequence practice
Sequences have their own daily drill: Number Sequence Practice. For both skills in one test, take the 60-question Sequences & Probability test.
Related practice
- Robot Games: timed Play/Pass EV decisions
- Quant interview brain teasers: worked probability & EV puzzles
- How to prepare for a quant trading interview: the full pipeline and a study plan
Questions
What kinds of questions are in the daily drill?
15 questions in 3 minutes: dice-sum probability, die-roll expected value, coin-flip runs, card draws, binary-bet EV, and "at least one" / "never" run probability. Answers accept decimals or fractions.
What's a good score on this drill?
11 of 15 is a reasonable target. This is a daily practice drill, so there is no firm cutoff or official pass mark. Keep first attempts separate from retakes, then try the longer Sequences & Probability practice test if you want a mixed timed session.
How do I get faster at EV questions specifically?
Write down each outcome × probability pair before summing. Under time pressure, most errors come from pairing an outcome with the wrong probability rather than from the arithmetic. For "at least one" questions, compute the complement (nothing happens at all) and subtract it from 1. That is almost always faster than adding up every success case.
Why is this separate from sequences now?
They are different skills: probability and EV calculation, and pattern recognition in a series. An older test combined them only because both are generic firm-test formats. This page covers probability and EV only. Number patterns have their own Daily Sequence drill.
Is there a longer, combined version?
Yes. The original 60-question Sequences & Probability test mixes both skills in one full-length fixed set. This daily drill is shorter, covers probability only and rotates every day.