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

Enable Like and Comment Medium | Strategy |

After the revolution, each of the 66 citizens of a certain city, including the king, has a salary of 1. King cannot vote, but has the power to suggest changes - namely, redistribution of salaries. Each person's salary must be a whole number of dollars, and the salaries must sum to 66. He suggests a new salary plan for every person including himelf in front of the city. Citizens are greedy, and vote yes if their salary is raised, no if decreased, and don't vote otherwise. The suggested plan will be implemented if the number of "yes" votes are more than "no" votes. The king is both, selfish and clever. He proposes a series of such plans. What is the maximum salary he can obtain for himself?

Hint

notice: (1) that the king must temporarily give up his own salary to get things started, and (2) that the game is to reduce the number of salaried citizens at each stage.

Answer

63

Solution

Continuing from hint, The king begins by proposing that 33 citizens have their salaries doubled to $2, at the expense of the remaining 33 (himself included). Next, he increases the salaries of 17 of the 33 salaried voters (to $3 or $4) while reducing the remaining 16 to $0. In successive turns, the number of salaried voters falls to 9, 5, 3, and 2. Finally, the king bribes three paupers with $1 each to help him turn over the two big salaries to himself, thus finishing with a royal salary of $63. It is not difficult to see that the king can do no better at any stage than to reduce the number of salaried voters to just over half the previous number; in particular, he can never achieve a unique salaried voter. Thus, he can do no better than $63 for himself, and the six rounds above are optimal

Source: P. Winkler

Enable Like and Comment Latest solved Puzzles

Color Switches Weird Sequences Intersecting Pillars Consecutive sums Scaling a Square Difficulty Level

© BRAINSTELLAR |