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 Medium | Discrete Maths |

At a party of N people, some have a symmetric friendship. Symmetric means that if A is friends with B, then B is in turn friends with A. Prove that there are at-least two people with same number of friends.

Solution

Source: Top Quant Interview

Enable Like and Comment Medium | Probability |

p and q are two points chosen at random between 0 & 1. What is the probability that the ratio p/q lies between 1 & 2?

Hint

Graph Shading. (Whenever we see two uniform random variables, we graph them up!)

Solution

Assume that the points are x & y. Create x-y graph, and our desired region is the area between lines y=x & y=2x. This region is 1/4th of the rest.

Source: Written Test

Enable Like and Comment Medium | Probability |

Roll a die, and you get paid what the dice shows. But if you want, you can request a second chance & roll the die again; get paid what the second roll shows instead of the first. What is the expected value?

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