expectation

How Many Circles Will You Create in Your Bowl of Ramen?

atypicalquant
March 27, 2023January 5, 2024
11 Comments

There are 100 noodles in your bowl of ramen. You select two of their ends randomly and connect them, then put the resulting -maybe deformed- noodle back in the bowl. You repeat the process until there are no ends left to join. On average, how many circles will you create? For simplicity, provide the answer approximated to the closest integer.

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Mastering the Number-Guessing Game: Strategies Beyond 50%

atypicalquant
September 19, 2022January 4, 2024

Two numbers are randomly chosen from the closed interval [0,1]. Afterward, you see the first number. You now must guess if the unseen number is greater or smaller.
Can you find a strategy with more than a 50% probability of winning?

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Expected Count in the ‘Sum Over One’ Game

atypicalquant
August 22, 2022January 4, 2024

You play a game where you sum independent identically distributed uniform random variables over the interval [0,1]. The first time the sum exceeds one, record the count of needed variables. What is the expected number of variables?

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Calculating Expected Distance to Center on a Disc

atypicalquant
June 27, 2022January 4, 2024

The expectation of distances on a disc: On a unit disc, what is the expected distance from a uniformly random point to its center?

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Drawing Cards for an Ace: Calculating Expectations

atypicalquant
May 2, 2022January 4, 2024

Drawing cards from a standard shuffled poker deck, how many cards do you expect to draw until you get an ace?

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Optimal Strategy in a Number Guessing Game

atypicalquant
December 20, 2021January 4, 2024
1 Comment

You play a number guessing game in which you must pick a real number x, between 0 and 924. At the same time, the number y is uniformly and randomly selected from the same range. If x is greater than y, then you have to pay the square of the difference between the two numbers. If y is greater than or equal to x, you pay double the difference.

What number should you choose initially?

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Expected Rolls to See Every Number on a Die

atypicalquant
July 4, 2021January 4, 2024

A fair 6-sided die is rolled until you get every number on the die. The order in which the numbers appear does not matter.

What is the expected number of die rolls?

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Expected Tosses for Three Consecutive Heads

atypicalquant
June 12, 2021January 4, 2024

A fair coin is tossed repeatedly until three consecutive tosses are heads. What is the expected number of coin tosses?

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Investment Strategy for a Highly Volatile Stock

atypicalquant
April 29, 2021January 4, 2024

For a given stock, you are certain that for the next 100 days, it will move either 10% up or 10% down each day. You can invest now, but if you choose to do so, you have to hold it for the entire 100 days. Would you do so?

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Expected Sum of Distances in a Triangle

atypicalquant
April 11, 2021January 4, 2024

In a triangle with sides 45, 60, respectively 75, what is the expected sum of distances from a point to its sides?

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