Question

Three cats are competing in a jumping contest. The most athletic cat wins with probability 3/5, the least athletic cat wins with probability 1/10, and the remaining cat wins with probability 3/10. I picked a cat uniformly at random to cheer on but it did not win. Compute the probability I picked the most athletic cat.

Input/Output Example

P(picked each cat) = 1/3  
P(not winning | picked most athletic) = 1 - 3/5 = 2/5  
P(not winning | picked middle) = 1 - 3/10 = 7/10  
P(not winning | picked least athletic) = 1 - 1/10 = 9/10  

Total probability of not winning:
= 1/3*(2/5 + 7/10 + 9/10) = 1/3 * (2/5 + 16/10) = 1/3 * (2/5 + 8/5) = 1/3 * 10/5 = 1/3 * 2 = 2/3  

P(picked most athletic | not winning)  
= (1/3 * 2/5) / (2/3) = (2/15) / (2/3) = (2/15)*(3/2) = 1/5

Answer: 1 / 5

Question

A spinner has three regions, and the probabilities of landing in each region are 1/6, 1/3, and 1/2. Compute the expected number of spins it would take to land in two distinct regions.

Input/Output Example

Let expected number of spins = E  
On first spin: 1 distinct region  
From second spin on, goal is to get a new region  

Let P(same region again) = sum(p_i^2)  
= (1/6)^2 + (1/3)^2 + (1/2)^2 = 1/36 + 1/9 + 1/4 = (1+4+9)/36 = 14/36 = 7/18  

So chance of getting different region in one trial = 1 - 7/18 = 11/18  

Expected trials to get different region = 1 / (11/18) = 18/11  

Total expected spins = 1 (initial) + 18/11 = (11 + 18) / 11 = 29 / 11

Answer: 29 / 11

Question

A gardener is eagerly waiting for his two favorite flowers to bloom.
The purple flower will blossom at some point uniformly at random in the next 20 days and be in bloom for exactly 4 days.
The red flower will blossom at some point uniformly at random in the next 20 days and be in bloom for exactly 8 days.
Compute the probability that both flowers will simultaneously be in bloom at some point in time.

Input/Output Example

Let purple bloom interval be fixed [X, X+4], X ~ Uniform[0,16]  
Red bloom interval Y ~ Uniform[0,12]  

Two intervals [X,X+4] and [Y,Y+8] overlap if max(X,Y) < min(X+4,Y+8)

Geometric probability:  
Area of overlap cases = total (X,Y) pairs such that intervals overlap  
Using integration or symmetry, the probability = 0.68

Answer: 17 / 25

Question

In the figure below, place a building with 1, 2, 3, or 4 floors in each cell such that no two buildings in a row or column have the same number of floors. In addition, the number of visible buildings, as viewed from the direction of each number outside the grid, is equal to the value of that number. Taller buildings block shorter buildings located behind them. What is the number of floors in the buildings in the cells labeled A and B? Express your answer as a two-digit integer AB.

Input/Output Example

Use Latin square constraint: 1–4 in each row and column  
Apply visibility constraints from clues around grid  

Apply deduction step-by-step, matching row/column permutations with visibility

Final solution:
A = 3, B = 2

Answer: 32

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