Probability interview questions
Probability is the core of quant trading and research interviews: expected value, conditioning, dice, cards, coins and random walks. Below are problems in the style interviewers ask, each with the answer and the method. Try each one before you open the solution.
1You roll a fair die until a 6 appears. What is the expected number of rolls?
Show the answer and solution
Answer: 6
Geometric distribution with p = 1/6, so E = 1/p = 6.
2You flip two fair coins. Given that at least one landed heads, what is the probability both did?
Show the answer and solution
Answer: 1/3
HH, HT and TH remain equally likely. Only one of the three is HH.
3Roll two fair dice. What is the probability the sum is 7?
Show the answer and solution
Answer: 1/6
Six of the 36 equally likely outcomes sum to 7.
4Ten cards are shuffled at random. What is the expected number that end up in their original position?
Show the answer and solution
Answer: 1
Each card stays put with probability 1/10, and there are 10 cards, so the expectation is 1.
5Roll a fair die four times. What is the probability of at least one 6? (Decimal is fine.)
Show the answer and solution
Answer: 671/1296 ≈ 0.5177
1 − (5/6)⁴ = 671/1296 ≈ 0.5177.
6X and Y are independent Uniform(0,1). What is E[max(X, Y)]?
Show the answer and solution
Answer: 2/3
The density of the max is 2t, so E = ∫ 2t² dt = 2/3. In general n/(n+1).
7You draw 2 cards from a standard 52-card deck. What is the probability both are aces?
Show the answer and solution
Answer: 1/221 ≈ 0.004525
(4/52) × (3/51) = 1/221.
8Three independent Uniform(0,1) draws. What is the expected value of the smallest?
Show the answer and solution
Answer: 1/4
E[min of n uniforms] = 1/(n+1) = 1/4.
9A bag holds 3 red and 2 blue balls. Draw 2 without replacement. What is the probability they are the same color?
Show the answer and solution
Answer: 2/5
(C(3,2) + C(2,2)) / C(5,2) = 4/10.
10Flip 10 fair coins. Given the first is heads, what is the expected number of heads?
Show the answer and solution
Answer: 11/2
1 + 9 × ½ = 5.5.
11Roll three fair dice. What is the probability all three show different numbers?
Show the answer and solution
Answer: 5/9
6 × 5 × 4 / 216 = 120/216 = 5/9.
12Roll two fair dice. What is the probability at least one shows a 6?
Show the answer and solution
Answer: 11/36 ≈ 0.3056
1 − (5/6)² = 11/36.
13Two independent fair dice. What is the expected value of their product?
Show the answer and solution
Answer: 49/4
E[XY] = E[X]E[Y] = 3.5 × 3.5 = 12.25.
14Flip a fair coin until tails. You get $1 for every head before the first tail. What is your expected payout?
Show the answer and solution
Answer: 1
E = (1 − p)/p with p = ½, which is 1.
15Three doors, one prize. You pick a door, the host opens a different empty door and offers a switch. What is your probability of winning if you switch?
Show the answer and solution
Answer: 2/3
Switching wins whenever the first pick was wrong, which happens with probability 2/3.
- Your first pick is right with probability ⅓ and wrong with probability ⅔.
- The host always opens an empty door you didn't pick, which gives no new information about your own door.
- If your first pick was wrong (⅔), the prize is behind the one remaining closed door, so switching wins.
- P(win by switching) = 2/3.
16A family has two children and the older one is a girl. What is the probability both are girls?
Show the answer and solution
Answer: 1/2
Only the younger child is unknown, so the probability is ½.
17Draw two cards from a shuffled deck. What is the probability they share a suit?
Show the answer and solution
Answer: 4/17 ≈ 0.2353
12 of the remaining 51 cards match its suit: 12/51 = 4/17.
18What is the variance of the number of heads in 100 fair coin flips?
Show the answer and solution
Answer: 25
100 × ½ × ½ = 25.
19Roll two fair dice. What is the probability their product is even?
Show the answer and solution
Answer: 3/4
1 − (½)² = 3/4.
20Z is standard normal. What is P(Z > 1.96), to three decimals?
Show the answer and solution
Answer: 1/40 ≈ 0.025
1.96 leaves 2.5% in each tail.
21You roll a fair 4-sided die until it shows one of 2 particular faces. What is the expected number of rolls?
Show the answer and solution
Answer: 2
Each roll succeeds with probability 2/4, so the wait is geometric with mean 4/2 = 2.
22You roll a fair 4-sided die 4 times. What is the probability that a 4 appears at least once? (Decimal is fine.)
Show the answer and solution
Answer: 175/256 ≈ 0.6836
It's easier to find the chance it never appears. Each roll misses with probability 3/4, and the rolls are independent, so P(never) = (3/4)^4 ≈ 0.3164. So P(at least once) = 1 − 0.3164 ≈ 0.6836.
23You roll two fair 4-sided dice. What is the probability the sum is exactly 5?
Show the answer and solution
Answer: 1/4
There are 4 × 4 = 16 equally likely ordered outcomes. The ones summing to 5 are (1,4), (2,3), (3,2), (4,1): 4 in total. So the probability is 4/16 = 1/4.
24You flip a fair coin 3 times. What is the probability of exactly 3 heads?
Show the answer and solution
Answer: 1/8
All 2^3 = 8 sequences of flips are equally likely. Choosing which 3 flips are heads gives C(3,3) = 1 of them. So the probability is 1/8.
25A bag holds 1 red and 4 blue balls. You draw 2 without replacement. What is the probability they are the same color?
Show the answer and solution
Answer: 3/5
There are C(5,2) = 10 equally likely pairs. Same color means both red (C(1,2) = 0 pairs) or both blue (C(4,2) = 6 pairs): (0 + 6)/10 = 3/5.
26A bag holds 2 red and 3 blue balls. You draw 2 without replacement. What is the probability both are red?
Show the answer and solution
Answer: 1/10
The first ball is red with probability 2/5. After that, 1 reds remain among 4 balls, so the second is red with probability 1/4. Multiply: 1/10.
27A coin lands heads with probability 0.3. You flip it 4 times. What is the probability of exactly 2 heads? (Decimal.)
Show the answer and solution
Answer: 1323/5000 ≈ 0.2646
Any one sequence with 2 heads and 2 tails has probability 0.3^2 × 0.7^2 ≈ 0.0441. There are C(4,2) = 6 such sequences, so the total is 6 × 0.0441 ≈ 0.2646.
28You have $2 and bet $1 on fair coin flips until you reach $5 or go broke. What is the probability you reach $5?
Show the answer and solution
Answer: 2/5
In a fair game your expected wealth stays at $2. You finish with $5 (probability p) or $0, so 5p = 2 and p = 2/5.
29You draw 2 independent Uniform(0,1) numbers. What is the expected value of the largest?
Show the answer and solution
Answer: 2/3
The max is below t only if all 2 draws are, so P(max ≤ t) = t^2 and its density is 2t^1. Then E[max] = ∫₀¹ t × 2t^1 dt = 2/3. (General rule: n/(n+1).)
30You draw 2 independent Uniform(0,1) numbers. What is the expected value of the smallest?
Show the answer and solution
Answer: 1/3
Flip every draw to 1 − X: the smallest becomes the largest. The largest of 2 uniforms averages 2/3, so the smallest averages 1 − 2/3 = 1/3.
31A fair die has 2 faces. What is the expected number of rolls to see every face at least once? (Decimal.)
Show the answer and solution
Answer: 3
Split the wait into stages. After seeing i faces, a new face appears with probability (2 − i)/2, so that stage takes 2/(2 − i) rolls on average. Add the stages: 2/2 + 2/1 + … + 2/1 = 2 × (1 + 1/2 + … + 1/2) ≈ 3.
32Each $1 bet wins with probability 0.4. Starting with $3, you play until you have $6 or nothing. What is the probability you reach $6? (Decimal.)
Show the answer and solution
Answer: 8/35 ≈ 0.2286
r = 1.5, so ((1.5)^3 − 1)/((1.5)^6 − 1) ≈ 0.2286.
- Let Pₖ be the chance of reaching $6 from $k. One bet later: Pₖ = 0.4·Pₖ₊₁ + 0.6·Pₖ₋₁, with P₀ = 0 and P6 = 1.
- Try Pₖ = A + B·rᵏ. Substituting shows it works when r = q/p = 0.6/0.4 = 1.5.
- The boundary conditions fix A and B: Pₖ = (rᵏ − 1)/(r^6 − 1).
- At k = 3: (1.5^3 − 1)/(1.5^6 − 1) = 2.375 / 10.39 ≈ 0.2286.