Probability Challenge: Find Interval of Integers Drawn from Urn

In summary, when an urn contains $n$ balls numbered $1, 2, . . . , n$ and they are drawn one at a time at random until the urn is empty, the probability of the numbers on the balls forming an interval of integers throughout the process is $\dfrac{2^{n-1}}{n!}$. This is calculated by considering the number of allowable processes starting with a specific first ball, which is $n-1\choose r-1$, and then summing over all possible starting balls. The total number of ways of drawing the balls is $n!$, making the probability of a process being allowable $\dfrac{2^{n-1}}{n!}$.
  • #1
lfdahl
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An urn contains $n$ balls numbered $1, 2, . . . , n$. They are drawn one at a time at random until the urn is empty.
Find the probability that throughout this process the numbers on the balls which have been drawn is an interval of integers.
(That is, for $1 \leq k \leq n$, after the $k$th draw the smallest number drawn equals the largest drawn minus $k − 1$.)
 
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  • #2
lfdahl said:
An urn contains $n$ balls numbered $1, 2, . . . , n$. They are drawn one at a time at random until the urn is empty.
Find the probability that throughout this process the numbers on the balls which have been drawn is an interval of integers.
(That is, for $1 \leq k \leq n$, after the $k$th draw the smallest number drawn equals the largest drawn minus $k − 1$.)
[sp]Call the process "allowable" if it satisfies that condition.

Suppose that the first ball drawn is numbered $r$. If the process is to be allowable then the number on each subsequent ball drawn must be next to either the lower end ($L$) or the upper end ($U$) of the existing consecutive run of integers. There are $n-1$ more balls to be drawn, so the process is completely specified by a string of $n-1$ letters $L$ and $U$. Also, there are exactly $r-1$ numbers less than $r$, so the string must contain $r-1$ $L$s (and $n-r$ $U$s). The number of such strings is \(\displaystyle n-1\choose r-1\). Therefore there are \(\displaystyle n-1\choose r-1\) allowable processes starting with $r$. So the total number of allowable processes is $$\sum_{r=1}^n{n-1\choose r-1} = 2^{n-1}.$$ The number of all ways of drawing the balls from the urn is $n!$. Therefore the probability of a process being allowable is $\dfrac{2^{n-1}}{n!}.$

[/sp]
 
  • #3
Opalg said:
[sp]Call the process "allowable" if it satisfies that condition.

Suppose that the first ball drawn is numbered $r$. If the process is to be allowable then the number on each subsequent ball drawn must be next to either the lower end ($L$) or the upper end ($U$) of the existing consecutive run of integers. There are $n-1$ more balls to be drawn, so the process is completely specified by a string of $n-1$ letters $L$ and $U$. Also, there are exactly $r-1$ numbers less than $r$, so the string must contain $r-1$ $L$s (and $n-r$ $U$s). The number of such strings is \(\displaystyle n-1\choose r-1\). Therefore there are \(\displaystyle n-1\choose r-1\) allowable processes starting with $r$. So the total number of allowable processes is $$\sum_{r=1}^n{n-1\choose r-1} = 2^{n-1}.$$ The number of all ways of drawing the balls from the urn is $n!$. Therefore the probability of a process being allowable is $\dfrac{2^{n-1}}{n!}.$

[/sp]

Awesome - thankyou for your participation, Opalg!
 

FAQ: Probability Challenge: Find Interval of Integers Drawn from Urn

1. What is the "Probability Challenge: Find Interval of Integers Drawn from Urn"?

The "Probability Challenge: Find Interval of Integers Drawn from Urn" is a mathematical problem where an urn contains a certain number of integers and the challenge is to determine the probability of drawing a specific interval of integers from the urn.

2. How does this challenge relate to probability?

This challenge relates to probability because it involves calculating the likelihood of drawing a specific interval of integers from a finite set of numbers. It requires an understanding of probability concepts such as sample space, events, and outcomes.

3. What factors affect the probability of drawing a specific interval of integers?

The factors that affect the probability of drawing a specific interval of integers include the total number of integers in the urn, the size of the interval, and whether the drawing is done with or without replacement.

4. What are the steps to solving this probability challenge?

The steps to solving this probability challenge are: 1) Identify the total number of integers in the urn, 2) Determine the size of the interval to be drawn, 3) Calculate the total number of possible outcomes, 4) Determine the number of favorable outcomes (intervals), 5) Divide the number of favorable outcomes by the total number of possible outcomes to get the probability.

5. Can this challenge be applied to real-life situations?

Yes, this challenge can be applied to real-life situations such as predicting the chances of winning a lottery or determining the probability of drawing a specific set of cards from a deck. It can also be used in statistical analysis and decision-making processes.

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