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

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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

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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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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

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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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