Compute answers to probability and counting puzzles
Company: Sig
Role: Data Scientist
Category: Statistics & Math
Difficulty: Hard
Interview Round: Technical Screen
Below are several independent interview-style math/logic questions.
## 1) Barn legs
You see only **spiders**, **chickens**, and **cows** in a barn.
- Total legs = **560**.
- #chickens = **2 ×** #cows.
- #spiders = **2 ×** #chickens.
How many spiders are there?
## 2) Seating toddlers (round table)
Five toddlers (Anna, Brian, Charlie, Dixie, Eva) sit around a **round** table (rotations considered the same).
Constraints:
1. Anna won’t sit next to Brian **or** Eva.
2. Brian won’t sit next to Charlie.
3. Dixie won’t sit next to Eva **or** Charlie.
Given **Dixie is sitting immediately to the right of Anna**, who is sitting immediately to the **left** of Brian?
## 3) Which factory? (Bayes)
Factory A produces 40% red, 60% black widgets.
Factory B produces 80% red, 20% black widgets.
- Choose a factory uniformly at random.
- Sample 2 widgets uniformly at random from that factory (independently).
Given both widgets are black, compute \(P(\text{Factory A} \mid \text{both black})\).
## 4) Dice game expected value
Roll three fair 6-sided dice.
- If all three are the same: win $20.
- If exactly two are the same: win $10.
- If all three are different: lose $2.
Compute the expected return per roll.
## 5) Bakery croissants
Each customer buys exactly one item.
- \(P(\text{croissant})=0.8\), \(P(\text{muffin})=0.2\).
- There are **2 croissants left**.
- **5 customers** remain in line.
Compute the probability the croissants are sufficient (i.e., at most 2 customers request a croissant).
## 6) Cats: truth/lie based on who has more
Asta (A), Bronya (B), and Clara (C) have **distinct** numbers of cats. Rule:
- When speaking to someone with **fewer** cats, the speaker tells the **truth**.
- When speaking to someone with **more** cats, the speaker **lies**.
Conversation:
1. B → C: “You have the most cats.”
2. A → B: “I have exactly 30% more cats than you.”
3. A → C: “Your cats are the average of mine and Bronya’s.” (\(C=(A+B)/2\))
4. C → A: “You have at least 4 more cats than me.” (\(A\ge C+4\))
How many cats does Bronya have?
## 7) Canoeing (relative speed)
River current flows at **2 mph** (constant). The two friends paddle at constant speed \(v\) mph **relative to the water**.
- Day 1: canoe **upstream for 4 hours**, then turn around and canoe **downstream for 5 hours** to the campsite.
- Day 2: canoe **back to the original starting point**, which is **23 miles upstream** from the campsite, and arrive at **16:00**.
What time did they leave on Day 2? (hh:mm)
## 8) Cookie arrangements
You baked:
- 4 indistinguishable snickerdoodles (S)
- 7 indistinguishable chocolate chip cookies (C)
How many distinct ways are there to arrange **5 cookies** in a line?
## 9) Cheering for a cat (conditional probability)
Three cats compete. Their win probabilities are:
- Most athletic wins with probability \(2/3\)
- Least athletic wins with probability \(1/12\)
- Remaining cat wins with probability \(1/4\)
You pick a cat uniformly at random to cheer for. You are told the cat you picked **did not win**.
Compute the probability you picked the most athletic cat.
## 10) Spinner: expected spins to see two regions
A spinner has 3 regions with landing probabilities \(1/2\), \(1/3\), and \(1/6\).
Compute the expected number of spins needed to have landed in **at least two distinct regions**.
## 11) Two blooming flowers (overlap probability)
Time is continuous in days. Each flower’s **start day** is uniform on \([0,20]\), independent.
- Purple blooms for exactly 4 days.
- Red blooms for exactly 8 days.
Compute the probability their bloom intervals overlap (i.e., they are simultaneously in bloom for a nonzero amount of time).
## 12) Expected number of pairs in a hand
A deck has 8 cards: two 10s, two Js, two Qs, two Ks.
You are dealt 4 cards without replacement.
Define a “pair” as having both cards of the same rank. Compute the expected number of pairs in the hand.
## 13) Bold betting strategy (reach 5 before 0)
You start with 3 tokens and want to reach 5 tokens before hitting 0.
Each turn, you bet as many tokens as possible but not more than needed to reach 5 (the “bold play” strategy).
You win each bet with probability \(2/3\); otherwise you lose.
Compute the probability of reaching 5 before 0.
## 14) Frog grid paths with a run-length constraint
A frog travels from \((0,0)\) to \((5,4)\).
- Each step is either 1 unit right (R) or 1 unit up (U).
- The frog refuses to make **three consecutive steps in the same direction** (no RRR and no UUU).
How many valid paths are there?
Quick Answer: This multi-part problem evaluates probabilistic reasoning, combinatorics and counting, expected-value computation, Bayes' inference, circular permutation constraints, algebraic Diophantine modeling, relative-speed calculations, and logical deduction under truth/lie rules in the Statistics & Math category for Data Scientist roles.