Compute answers to probability and counting puzzles

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?