probabilities

Non-Markovian Paper Toss Challenge: Aiming for 298 Wins

atypicalquant
February 20, 2023January 5, 2024

Let’s say you toss paper 300 times into a bin. You miss the first but nail the second one. Subsequently, your chance of hitting the bin is your cumulative win rate. What is the probability that you hit the can 298 times?

Decoding Probabilities: The Minimum Value of P(X)=0

atypicalquant
January 16, 2023January 4, 2024
1 Comment

X is a random variable, taking positive integer values and having: E[X] = 1, E[X^2] = 2, and E[X^3] = 5. What is the smallest value that the probability of X equals 0 can take?

Probability Analysis: Comparing X to 4Y in Random Variables

atypicalquant
May 30, 2022January 4, 2024

Solve the following random variable comparison: what is the probability that X > 4Y if :

1. X, Y independent, standard normally distributed?

2. X, Y independent, uniformly distributed on [0,1]?

Plane Boarding Puzzle: Will the Last Person Get Their Seat?

atypicalquant
February 7, 2022January 4, 2024

There are 100 people in line to board a plane that seats all of them. The first person in line is drunk and decides to take a random seat. Every person that boards the plane after them will either: take the seat on their ticket or if that seat is taken, a random one instead.

What is the probability that the last person that boards will end up in their assigned seat?

Coin Flip Game: Can You Win with Fewer Tosses?

atypicalquant
January 10, 2022January 4, 2024

Rick and Morty play a game where they flip 20 and 21 coins, respectively. What is the probability that Rick gets more heads?

Squid Game Bridge: Calculating Survival Odds

atypicalquant
November 2, 2021January 4, 2024

Ahead of the players is the Squid Game Bridge with 18 steps, each made of a pair of glass tiles: one of them can hold the weight of a player, while the other can’t.

Players advance on the bridge in a predetermined order, one pair of tiles at a time.

If they choose a solid tile, they advance, otherwise, they are eliminated.

What is the expected number of survivors in a 16-player game?

Calculating Odds for Third Player’s Tennis Win

atypicalquant
September 13, 2021January 4, 2024

Three equally skilled tennis players play in a tournament.

After each round, the winner plays against the third player.

The first to reach two consecutive wins becomes the champion.

If A and B play the first game, what is the probability that C is the tennis tournament winner?

Grid Path Probability: Through the Center to Cheese

atypicalquant
May 23, 2021January 4, 2024

A mouse starts at the bottom left corner of a $8 \times 6$ grid. At each step, he can only move RIGHT or UP, moving on the grid lines. He wants to get to the upper right corner, where the cheese is located.

probabilities

Let’s say you toss paper 300 times into a bin. You miss the first but nail the second one. Subsequently, your chance of hitting the bin is your cumulative win rate. What is the probability that you hit the can 298 times?

X is a random variable, taking positive integer values and having: E[X] = 1, E[X^2] = 2, and E[X^3] = 5. What is the smallest value that the probability of X equals 0 can take?

Solve the following random variable comparison: what is the probability that X > 4Y if :

1. X, Y independent, standard normally distributed?

2. X, Y independent, uniformly distributed on [0,1]?

There are 100 people in line to board a plane that seats all of them. The first person in line is drunk and decides to take a random seat. Every person that boards the plane after them will either: take the seat on their ticket or if that seat is taken, a random one instead.

What is the probability that the last person that boards will end up in their assigned seat?

Rick and Morty play a game where they flip 20 and 21 coins, respectively. What is the probability that Rick gets more heads?

Ahead of the players is the Squid Game Bridge with 18 steps, each made of a pair of glass tiles: one of them can hold the weight of a player, while the other can’t.

Players advance on the bridge in a predetermined order, one pair of tiles at a time.

If they choose a solid tile, they advance, otherwise, they are eliminated.

What is the expected number of survivors in a 16-player game?

Three equally skilled tennis players play in a tournament.

After each round, the winner plays against the third player.

The first to reach two consecutive wins becomes the champion.

If A and B play the first game, what is the probability that C is the tennis tournament winner?

A mouse starts at the bottom left corner of a grid. At each step, he can only move RIGHT or UP, moving on the grid lines. He wants to get to the upper right corner, where the cheese is located.

All unique paths are equally likely to be chosen. What’s the probability that he will pass through the center of the grid?

A mouse starts at the bottom left corner of a $8 \times 6$ grid. At each step, he can only move RIGHT or UP, moving on the grid lines. He wants to get to the upper right corner, where the cheese is located.