Q: You have a shuffled deck of 52 cards. You turn over cards one at a time. I'll pay you $1 for each red card. You can stop at any time and keep your earnings, but if you turn over all 52 cards you get $26. What is the optimal strategy and expected value?

$26. No matter when you stop, the expected payout is always $26. At any point, the conditional expected value of the remaining cards is always proportional — the fraction of red cards remaining always has the same expectation. So every stopping strategy yields the same expected value of $26.

Q: A drunk man starts at the origin on a number line. Each step he goes +1 or -1 with equal probability. What is the expected number of steps to reach +1 or -1 from the origin?

1. From the origin, the very first step takes him to either +1 or -1 with certainty. So the expected number of steps is exactly 1.

Q: You have 10 pairs of socks in a drawer (20 socks total), each pair a different color. You grab socks one at a time in the dark. What is the expected number of socks you need to grab to get a matching pair?

By the pigeonhole principle, you're guaranteed a match after at most 11 socks. For the expected value: the probability of no match after k draws is (20/20)(18/19)(16/18)..., using a birthday-problem-style calculation. The expected number is approximately 5.7 socks.

Q: 100 passengers board a plane with 100 assigned seats. The first passenger sits randomly. Each subsequent passenger sits in their seat if available, otherwise picks randomly. What is the probability the 100th passenger sits in their assigned seat?

1/2. By symmetry, when the last passenger boards, only seat 1 or seat 100 can be empty. Every random choice treats these two seats with equal probability, so there is a 50/50 chance.

Q: What is the expected number of times you need to roll a fair die to see all 6 faces?

14.7. This is the coupon collector's problem. After seeing k distinct faces, the probability of seeing a new face is (6-k)/6, so the expected rolls for the next new face is 6/(6-k). Sum: 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 1 + 1.2 + 1.5 + 2 + 3 + 6 = 14.7.

Q: If you roll two fair six-sided dice, what is the probability the sum is exactly 7?

1/6. There are 36 total outcomes. The combinations summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — that's 6 outcomes. 6/36 = 1/6.

Q: When rolling two fair six-sided dice, what is the probability that at least one shows a 5?

11/36. The complement (no 5s) has probability (5/6)^2 = 25/36. So P(at least one 5) = 1 - 25/36 = 11/36.

Q: A family has two children. At least one is a girl. What is the probability both are girls?

1/3. Possible combinations: BB, BG, GB, GG. Eliminating BB leaves BG, GB, GG. Only GG has both girls, so 1/3.

Question 1

You flip a fair coin until you get two heads in a row. What is the expected number of flips?

Accepted Answer

6. Let E be the expected number of flips. From the start, you flip once (cost 1). If tails (prob 1/2), you restart. If heads (prob 1/2), you flip again. If the second is heads, you're done (total cost so far: 2 from this branch). If tails, you restart. Setting up the equation: E = 1 + (1/2)E + (1/2)(1 + (1/2)(0) + (1/2)E) simplifies to E = 6.

Question 2

100 prisoners are each assigned a unique number 1-100. Boxes labeled 1-100 each contain a random number. Each prisoner may open 50 boxes. They all go free if every prisoner finds their own number. They can strategize beforehand but not communicate after. What strategy maximizes survival probability?

Accepted Answer

Each prisoner starts by opening the box matching their number, then follows the chain (opens the box with the number they found). They succeed if their chain length is <= 50. The probability that the longest cycle in a random permutation exceeds 50 is about ln(2) = 0.69, so the survival probability is about 31%. This is vastly better than the (1/2)^100 probability from random guessing.

Question 3

You have a shuffled deck of 52 cards. You turn over cards one at a time. I'll pay you $1 for each red card. You can stop at any time and keep your earnings, but if you turn over all 52 cards you get $26. What is the optimal strategy and expected value?

Accepted Answer

$26. No matter when you stop, the expected payout is always $26. At any point, the conditional expected value of the remaining cards is always proportional — the fraction of red cards remaining always has the same expectation. So every stopping strategy yields the same expected value of $26.

Question 4

A drunk man starts at the origin on a number line. Each step he goes +1 or -1 with equal probability. What is the expected number of steps to reach +1 or -1 from the origin?

Accepted Answer

1. From the origin, the very first step takes him to either +1 or -1 with certainty. So the expected number of steps is exactly 1.

Question 5

You have 10 pairs of socks in a drawer (20 socks total), each pair a different color. You grab socks one at a time in the dark. What is the expected number of socks you need to grab to get a matching pair?

Accepted Answer

By the pigeonhole principle, you're guaranteed a match after at most 11 socks. For the expected value: the probability of no match after k draws is (20/20)(18/19)(16/18)..., using a birthday-problem-style calculation. The expected number is approximately 5.7 socks.

Question 6

100 passengers board a plane with 100 assigned seats. The first passenger sits randomly. Each subsequent passenger sits in their seat if available, otherwise picks randomly. What is the probability the 100th passenger sits in their assigned seat?

Accepted Answer

1/2. By symmetry, when the last passenger boards, only seat 1 or seat 100 can be empty. Every random choice treats these two seats with equal probability, so there is a 50/50 chance.

Question 7

What is the expected number of times you need to roll a fair die to see all 6 faces?

Accepted Answer

14.7. This is the coupon collector's problem. After seeing k distinct faces, the probability of seeing a new face is (6-k)/6, so the expected rolls for the next new face is 6/(6-k). Sum: 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 1 + 1.2 + 1.5 + 2 + 3 + 6 = 14.7.

Question 8

If you roll two fair six-sided dice, what is the probability the sum is exactly 7?

Accepted Answer

1/6. There are 36 total outcomes. The combinations summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — that's 6 outcomes. 6/36 = 1/6.

Question 9

When rolling two fair six-sided dice, what is the probability that at least one shows a 5?

Accepted Answer

11/36. The complement (no 5s) has probability (5/6)^2 = 25/36. So P(at least one 5) = 1 - 25/36 = 11/36.

Question 10

A family has two children. At least one is a girl. What is the probability both are girls?

Accepted Answer

1/3. Possible combinations: BB, BG, GB, GG. Eliminating BB leaves BG, GB, GG. Only GG has both girls, so 1/3.

Question 11

What is the expected value of a single roll of a fair six-sided die?

Accepted Answer

3.5. E = (1+2+3+4+5+6)/6 = 21/6 = 3.5.

Question 12

Drawing two cards from a 52-card deck without replacement, what is the probability both are Aces?

Accepted Answer

1/221. P = (4/52) x (3/51) = 12/2652 = 1/221.

Question 13

In a room of 23 people, what is the approximate probability that at least two share the same birthday?

Accepted Answer

Approximately 50.7%. Calculate P(no match) = (365/365)(364/365)(363/365)...(343/365) ≈ 0.493. So P(match) ≈ 1 - 0.493 = 0.507.

Question 14

If you roll 10 fair six-sided dice, what is the probability that the sum is divisible by 6?

Accepted Answer

1/6. No matter what the first 9 dice show, exactly one value on the 10th die (from 1-6) makes the total divisible by 6. Each face is equally likely, so P = 1/6.

Question 15

You are on a game show with three doors. Behind one is a car; behind the others, goats. You pick Door 1. The host opens Door 3 (a goat). Should you switch to Door 2?

Accepted Answer

Yes, you should always switch. The key insight is that the host knows where the car is and will always reveal a goat. When you first pick, you have a 1/3 chance of being right. The host then deliberately opens a door he knows has a goat, which concentrates the remaining 2/3 probability onto the other unopened door. Switching wins whenever your initial pick was wrong (2/3 of the time). If the host had opened a door randomly and happened to show a goat, switching would not help — it is the host's knowledge that creates the advantage.

Question 16

Three ants sit at the corners of an equilateral triangle. Each randomly picks clockwise or counter-clockwise and walks along the edge. What is the probability none collide?

Accepted Answer

1/4. There are 2^3 = 8 direction combinations. No collision only if all go the same way (all CW or all CCW). That's 2/8 = 1/4.

Question 17

You flip a fair coin 17 times, your friend flips 16 times. What is the probability you flip more heads?

Accepted Answer

1/2. After 16 flips each, by symmetry P(you lead) = P(friend leads). The 17th flip breaks ties in your favor or maintains your lead, resulting in exactly 50%.

Question 18

If you roll three fair six-sided dice, what is the probability the numbers are in strictly increasing order?

Accepted Answer

5/54. There are 216 total outcomes. Choose 3 distinct numbers from 6: C(6,3) = 20 ways, each with exactly one increasing arrangement. P = 20/216 = 5/54.

Question 19

What is the probability of drawing two cards of the same suit from a 52-card deck without replacement?

Accepted Answer

4/17. The first card can be anything. Then 12 of the remaining 51 cards match its suit. P = 12/51 = 4/17.

Question 20

You call three friends in Seattle, each tells the truth with probability 2/3. All three say it's raining. If the base probability of rain is 10%, what is the actual probability it's raining?

Accepted Answer

About 47%. Using Bayes: P(3 Yes|Rain) = (2/3)^3 = 8/27. P(3 Yes|No Rain) = (1/3)^3 = 1/27. P(Rain|3 Yes) = (8/27 × 0.1) / (8/27 × 0.1 + 1/27 × 0.9) = 8/17 ≈ 47%.

Question 21

On average, how many times must you roll a fair die before seeing a 6?

Accepted Answer

6. This is a geometric distribution with p = 1/6. Expected value = 1/p = 6.

Question 22

In tennis, you win each point with probability 60%. What is the probability of winning from Deuce?

Accepted Answer

About 69%. Let P = probability of winning from deuce. P = W^2 / (1 - 2W(1-W)) where W = 0.6. P = 0.36/0.52 ≈ 0.692.

Question 23

What is the probability of rolling a sum of 11 with two fair six-sided dice?

Accepted Answer

1/18. Only (5,6) and (6,5) sum to 11. That's 2/36 = 1/18.

Question 24

What is the probability of drawing a red card from a standard 52-card deck?

Accepted Answer

1/2. Hearts and Diamonds are red (26 cards). 26/52 = 1/2.

Question 25

In a room of 10 people, if everyone shakes hands with everyone else exactly once, how many handshakes occur?

Accepted Answer

45. C(10,2) = 10 × 9 / 2 = 45.

Question 26

Four random points are chosen on a circle to form two chords. What is the probability the two chords intersect?

Accepted Answer

1/3. Four points can be paired into two chords in 3 ways. Only one pairing results in crossing chords. P = 1/3.

Question 27

What is the probability of getting an even number of heads when flipping a fair coin 100 times?

Accepted Answer

1/2. For any outcome of the first 99 flips, the 100th flip determines whether the total is even or odd with equal probability.

Question 28

An urn has 3 red, 4 blue, and 5 green balls. Drawing with replacement, what is the probability of drawing red before blue?

Accepted Answer

3/7. Green balls are irrelevant — we only care about when a red or blue is drawn. Among the 7 non-green balls, 3 are red. P = 3/7.

Question 29

Pick a random integer from 1 to 10. What is the probability it is prime?

Accepted Answer

2/5. Primes between 1 and 10: 2, 3, 5, 7. That's 4 out of 10 = 2/5.

Question 30

You have a fair coin and a double-headed coin. You pick one at random and flip it 3 times, getting 3 heads. What is the probability you picked the fair coin?

Accepted Answer

1/9. By Bayes: P(Fair|3H) = P(3H|Fair)P(Fair) / P(3H) = (1/8 × 1/2) / (1/8 × 1/2 + 1 × 1/2) = (1/16) / (9/16) = 1/9.

Question 31

A bag has 5 red and 5 blue marbles. Drawing two without replacement, what is the probability they are different colors?

Accepted Answer

5/9. Ways to pick one of each: 5 × 5 = 25. Total ways to pick 2: C(10,2) = 45. P = 25/45 = 5/9.

Question 32

What is the probability of not rolling a 6 in three consecutive rolls of a fair die?

Accepted Answer

125/216. P(not 6) per roll = 5/6. For three independent rolls: (5/6)^3 = 125/216.

Question 33

You pick three random numbers from Uniform(0,1). What is the probability the median exceeds 0.5?

Accepted Answer

1/2. By symmetry, the median is equally likely to be above or below 0.5.

Question 34

What is the probability a random person shares your exact birthday?

Accepted Answer

1/365. Assuming 365 equally likely days, exactly one matches yours.

Question 35

A stick is broken at two random points uniformly distributed along its length. What is the probability the three pieces can form a triangle?

Accepted Answer

1/4. For three lengths to form a triangle, each piece must be less than half the total length. If the stick has length 1 and we break at points X and Y (uniform on [0,1]), the condition is that all three pieces are < 1/2. The favorable region in the unit square has area 1/4.

Question 36

What is the probability that the sum of two fair dice is even?

Accepted Answer

1/2. Sum is even if both are even or both are odd. P = (1/2)(1/2) + (1/2)(1/2) = 1/2.

Question 37

What is the probability of drawing a King or Queen from a standard 52-card deck?

Accepted Answer

2/13. There are 4 Kings and 4 Queens = 8 cards. P = 8/52 = 2/13.

Question 38

If you flip a coin 3 times, what is the probability of getting heads on all three flips?

Accepted Answer

1/8. Each flip has P(H) = 1/2. For three independent flips: (1/2)^3 = 1/8.

Question 39

An event has a 20% chance of occurring. What is the probability it happens at least once in 5 trials?

Accepted Answer

About 67.2%. P(at least once) = 1 - P(never) = 1 - (0.8)^5 = 1 - 0.328 = 0.672.

Question 40

What is the probability that two fair six-sided dice show the same number?

Accepted Answer

1/6. The first die can be anything. The second must match, with probability 1/6.

Question 41

If you pick a random letter from 'PROBABILITY', what is the probability it is a vowel?

Accepted Answer

4/11. The vowels are O, A, I, I — 4 vowels out of 11 letters.

Question 42

What is the probability of drawing a face card (J, Q, K) from a standard deck?

Accepted Answer

3/13. There are 12 face cards in 52 cards. P = 12/52 = 3/13.

Question 43

What is the probability that the sum of two fair dice is a prime number?

Accepted Answer

5/12. Prime sums: 2(1 way), 3(2), 5(4), 7(6), 11(2) = 15 ways out of 36. P = 15/36 = 5/12.

Question 44

10 people enter an elevator in a 20-floor building. Each gets off at a random floor (2-20). What is the expected number of stops?

Accepted Answer

About 7.8. P(a specific floor is skipped) = (18/19)^10. P(stopped at) = 1 - (18/19)^10. Expected stops = 19 × (1 - (18/19)^10) ≈ 7.8.

Question 45

A bag has 3 fair coins and 1 double-headed coin. You pick one and flip it, getting heads. What is the probability you picked a fair coin?

Accepted Answer

3/5 (60%). By Bayes: P(Fair|H) = (0.5 × 0.75) / (0.5 × 0.75 + 1 × 0.25) = 0.375/0.625 = 0.6.

Question 46

For a 3 sets tennis game, would you bet on it finishing in 2 sets or 3 sets?

Accepted Answer

Two sets. Let p=prob team 1 wins and q=prob team 2 wins. p² + q² = probability finish in two sets. 2pq = probability finish in three sets. p² + q² is always >= 2pq, so the answer is two sets.

Question 47

I have a square, and place three dots along the 4 edges at random. What is the probability that the dots lie on distinct edges?

Accepted Answer

3/8. Given the edge the first dot is on, the probability the other two dots are on distinct edges is (3/4) × (2/4) = 6/16 = 3/8.

Question 48

Two decks of cards. One deck has 52 cards, the other has 104. You pick two cards separately from the same pack. If both cards are red, you win. Which pack will you choose?

Accepted Answer

104 card pack. (52/104) × (51/103) > (26/52) × (25/51), or equivalently 51/103 > 25/51.

Question 49

You have 17 coins and I have 16 coins. We flip all coins at the same time. If you have more heads you win, otherwise I win. What's your probability of winning?

Accepted Answer

0.5. The first 16 flips have the same probability distribution for both players. The game depends entirely on whether the 17th coin flip is heads or tails (50/50).

Question 50

What is the probability you draw two cards of the same color from a standard 52-card deck without replacement?

Accepted Answer

25/51. You draw any card first. Then there are 51 cards left and 25 of the same color. P = 25/51.

Question 51

In the World Series, what are the odds it goes 7 games if each team has an equal chance of winning?

Accepted Answer

20/64 = 5/16. In the first 6 games each team must win exactly 3. C(6,3) × (0.5)^6 = 20/64.

Question 52

Given 100 coin flips, what is the probability that you get an even number of heads?

Accepted Answer

1/2. Whether there is an odd or even number of heads is determined by the final flip (50/50 chance), for any number of flips.

Question 53

There are 5 balls, 3 red and 2 black. What is the probability that a random ordering of the 5 balls does not have the 2 black balls next to each other?

Accepted Answer

0.6. There are C(5,2) = 10 ways to place the black balls. 4 of those have them adjacent. So 6/10 = 0.6 do NOT have them next to each other.

Question 54

2 blue and 2 red balls in a box, no replacing. Guess the color of the ball, you receive a dollar if you are correct. What is the dollar amount you would pay to play this game?

Accepted Answer

$17/6 ≈ $2.83. You always get the last ball right ($1). First ball: 50% chance = $0.50. Second ball: given you know what was drawn, 2/3 chance of guessing right = $0.67. Third ball: 1/2 or certainty depending on prior draws. Expected = 1/2 + 2/3 + 5/6 + 1 = 17/6.

Question 55

Three people each pick a random number between 0 and 1. What is the probability that the sum is less than 1?

Accepted Answer

1/6. The sample space is the unit cube [0,1]^3 with volume 1. The favorable region is the tetrahedron where x + y + z < 1, which has volume (1/3!) = 1/6.

Quantitative Finance

Quantitative Finance

Quantitative Finance

Quantitative Finance