Medium | Strategy |

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied.

1) A new customer wants to check in, how will you accommodate her?

2) What if infinite number of people want to check in, how will you accommodate them?

3) Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?

1) A new customer wants to check in, how will you accommodate her?

2) What if infinite number of people want to check in, how will you accommodate them?

3) Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?

Hint

Define Infinity ;)

Solution

1) Since there are infinite number of rooms and infinite+1= infinite

Just ask person in room k to move to k+1, thus making the first room vacant. :)

2) In the other case, since infinite+infinite = infinite

asking person in room k to move to 2k solves the problem.

3) Since NxN is countable set. We can get a 1-1 mapping from N to NxN

Hence, we can accommodate (infinite people X infinite buses) in the hotel.

Relevant article:

http://en.wikipedia.org/wiki/Cantor_pairing_function

Just ask person in room k to move to k+1, thus making the first room vacant. :)

2) In the other case, since infinite+infinite = infinite

asking person in room k to move to 2k solves the problem.

3) Since NxN is countable set. We can get a 1-1 mapping from N to NxN

Hence, we can accommodate (infinite people X infinite buses) in the hotel.

Relevant article:

http://en.wikipedia.org/wiki/Cantor_pairing_function

Source: CSEblog

Enable Like and Comment Medium | Probability |

In this gambling game, a player can buy a ticket for Rs 1 on any number from 1 to 6. Three identical and unfair dice are rolled. If the booked number appears on 0, 1, 2 or 3 dice, player wins Rs 0, 1, 2 or 3 respectively, without returning the original Rs 1. What is expected money you can win after buying a ticket for Rs 1?

Hint

Book the full house

Answer

1/2

Solution

We bet Rs 1 on each number 1-6. In any case, we get Rs 3 back. That is 1/2 per ticket. Hence the expected amount of money we can win is 1/2. A tedious way to arrive at this answer is to calculate the probability of getting 1, 2 or 3 faces common to our booking.

Source: 50 puzzles in prob

Enable Like and Comment Medium | Discrete Maths |

A group has 70 members. For any two members X and Y there is a language that X speaks but Y does not, and there is a language that Y speaks but X does not. At least how many different languages are spoken by the members of this group?

Solution

8 choose 4 is 70

Source: Quantnet

Enable Like and Comment Medium | Probability |

A and B are in a team called AB, playing against C. If AB team wins it gets Rs 3, nothing otherwise.

Game is: A and B are placed in 2 separate rooms far away. A will toss a coin and B will also toss a coin; A will have to guess outcome of B's toss and B will guess A's. If both guesses are right, team AB wins Rs 3, nothing otherwise.

Should they play the game, by giving Rs 1 in start to C.

Game is: A and B are placed in 2 separate rooms far away. A will toss a coin and B will also toss a coin; A will have to guess outcome of B's toss and B will guess A's. If both guesses are right, team AB wins Rs 3, nothing otherwise.

Should they play the game, by giving Rs 1 in start to C.

Hint

Winning probability in not 1/4. They can make strategy before game.

Answer

1/2

Solution

They will have same coin with probability 1/2. They can speak their own coin's face as the guess of other's. They win game with probability 1/2. Pay off will be positive, and hence they should play!

Source: Top Quant Interview

Enable Like and Comment Medium | Probability |

Snow-particles are falling on the ground one after another. A particular snowflake turns out to be of type "Stellar Dendrite" with probability 'p' if its previous particle was also Stellar Dendrite, and with probability 'q' if previous one was something else. If a snowflake is picked from ground, what is the probability that it is Stellar Dendrite?

PS:Although no two snowflakes are alike, yet there are various crystalline structures to categorize their interesting shapes. The image depicts the most popular shape, called Stellar Dendrites, which means star-like particles with tree-like branches.

PS:Although no two snowflakes are alike, yet there are various crystalline structures to categorize their interesting shapes. The image depicts the most popular shape, called Stellar Dendrites, which means star-like particles with tree-like branches.

Hint

Need to form a recursive equation of conditional probability

Answer

probability is q/(1-p+q)

Solution

Solution by Palak:

Let x be the probability that a snowflake picked from ground is Stellar Dendrite. Thus, when a new snowflake is falling, with prob=x the last snowflake was Stellar Dendrite => prob the new falling snowflake is Stellar Dendrite = x*p + (1-x)*q. But, for the composition of the snowflakes on the ground to remain constant, xp+(1-x)q should be =x => x=1/(1+(1-p)/q)

This is a kind of steady state analysis.

Let x be the probability that a snowflake picked from ground is Stellar Dendrite. Thus, when a new snowflake is falling, with prob=x the last snowflake was Stellar Dendrite => prob the new falling snowflake is Stellar Dendrite = x*p + (1-x)*q. But, for the composition of the snowflakes on the ground to remain constant, xp+(1-x)q should be =x => x=1/(1+(1-p)/q)

This is a kind of steady state analysis.

Source: Self

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