On the real line place an object at 1. After every flip of a fair coin move the object to the right by 1 unit...












1












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On the real line place an object at 1. After every flip of a fair coin move the object to the right by 1 unit if the outcome is head and to the left by 1 unit if the outcome is tail.



Let $N$ be a fixed positive integer.Game ends when the object reaches either $0$ or $N$.



What is the probability of the object reaching at $N$, i.e $P(N)$.
Give me some hint..










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closed as off-topic by Saad, Arnaud D., Xander Henderson, Did, user21820 Jan 14 at 15:05


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Arnaud D., Xander Henderson, Did, user21820

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 1




    $begingroup$
    This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
    $endgroup$
    – jmerry
    Jan 12 at 8:08
















1












$begingroup$


On the real line place an object at 1. After every flip of a fair coin move the object to the right by 1 unit if the outcome is head and to the left by 1 unit if the outcome is tail.



Let $N$ be a fixed positive integer.Game ends when the object reaches either $0$ or $N$.



What is the probability of the object reaching at $N$, i.e $P(N)$.
Give me some hint..










share|cite|improve this question









$endgroup$



closed as off-topic by Saad, Arnaud D., Xander Henderson, Did, user21820 Jan 14 at 15:05


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Arnaud D., Xander Henderson, Did, user21820

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 1




    $begingroup$
    This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
    $endgroup$
    – jmerry
    Jan 12 at 8:08














1












1








1


3



$begingroup$


On the real line place an object at 1. After every flip of a fair coin move the object to the right by 1 unit if the outcome is head and to the left by 1 unit if the outcome is tail.



Let $N$ be a fixed positive integer.Game ends when the object reaches either $0$ or $N$.



What is the probability of the object reaching at $N$, i.e $P(N)$.
Give me some hint..










share|cite|improve this question









$endgroup$




On the real line place an object at 1. After every flip of a fair coin move the object to the right by 1 unit if the outcome is head and to the left by 1 unit if the outcome is tail.



Let $N$ be a fixed positive integer.Game ends when the object reaches either $0$ or $N$.



What is the probability of the object reaching at $N$, i.e $P(N)$.
Give me some hint..







probability combinatorics






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asked Jan 12 at 7:55









Supriyo BanerjeeSupriyo Banerjee

1176




1176




closed as off-topic by Saad, Arnaud D., Xander Henderson, Did, user21820 Jan 14 at 15:05


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Arnaud D., Xander Henderson, Did, user21820

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by Saad, Arnaud D., Xander Henderson, Did, user21820 Jan 14 at 15:05


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Arnaud D., Xander Henderson, Did, user21820

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    $begingroup$
    This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
    $endgroup$
    – jmerry
    Jan 12 at 8:08














  • 1




    $begingroup$
    This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
    $endgroup$
    – jmerry
    Jan 12 at 8:08








1




1




$begingroup$
This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
$endgroup$
– jmerry
Jan 12 at 8:08




$begingroup$
This is a case of the famous "gambler's ruin" problem. For this version with a fair game, looking at the probability of reaching one state before another, the easiest approach is to track expected values. OK, where's a good version of this to link to on MSE?
$endgroup$
– jmerry
Jan 12 at 8:08










1 Answer
1






active

oldest

votes


















6












$begingroup$

Here is a short proof using induction and symmetry. We use induction on $N$ to show that $P(N) = frac{1}{N}$. For $N = 1$, this is obvious.



Now, suppose that $P(k) = frac{1}{k}$ for $k leq n$. We shall show that the statement holds for $N = n+1$. Note that, in order to reach $n+1$, we must reach $n$. The probability of this is $P(n) = frac{1}{n}$. Now, the probability of reaching $n+1$ when standing on $n$ is the same as the probability of reaching $0$ when standing on $1$, by symmetry. So we get
$$P(n+1) = P(n)(1 - P(n+1)) Leftrightarrow P(n+1) = frac{P(n)}{1+P(n)} = frac{frac{1}{n}}{frac{n+1}{n}} = frac{1}{n+1}$$
So by the principle of induction, $P(N) = frac{1}{N}$ for all positive integers $N$






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  • $begingroup$
    +1 Nice solution.
    $endgroup$
    – drhab
    Jan 12 at 9:26










  • $begingroup$
    Good answer, but you switches from k to n there
    $endgroup$
    – DreamConspiracy
    Jan 12 at 12:56










  • $begingroup$
    Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
    $endgroup$
    – Pedro A
    Jan 12 at 13:13












  • $begingroup$
    The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
    $endgroup$
    – nesHan
    Jan 12 at 14:01


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









6












$begingroup$

Here is a short proof using induction and symmetry. We use induction on $N$ to show that $P(N) = frac{1}{N}$. For $N = 1$, this is obvious.



Now, suppose that $P(k) = frac{1}{k}$ for $k leq n$. We shall show that the statement holds for $N = n+1$. Note that, in order to reach $n+1$, we must reach $n$. The probability of this is $P(n) = frac{1}{n}$. Now, the probability of reaching $n+1$ when standing on $n$ is the same as the probability of reaching $0$ when standing on $1$, by symmetry. So we get
$$P(n+1) = P(n)(1 - P(n+1)) Leftrightarrow P(n+1) = frac{P(n)}{1+P(n)} = frac{frac{1}{n}}{frac{n+1}{n}} = frac{1}{n+1}$$
So by the principle of induction, $P(N) = frac{1}{N}$ for all positive integers $N$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1 Nice solution.
    $endgroup$
    – drhab
    Jan 12 at 9:26










  • $begingroup$
    Good answer, but you switches from k to n there
    $endgroup$
    – DreamConspiracy
    Jan 12 at 12:56










  • $begingroup$
    Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
    $endgroup$
    – Pedro A
    Jan 12 at 13:13












  • $begingroup$
    The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
    $endgroup$
    – nesHan
    Jan 12 at 14:01
















6












$begingroup$

Here is a short proof using induction and symmetry. We use induction on $N$ to show that $P(N) = frac{1}{N}$. For $N = 1$, this is obvious.



Now, suppose that $P(k) = frac{1}{k}$ for $k leq n$. We shall show that the statement holds for $N = n+1$. Note that, in order to reach $n+1$, we must reach $n$. The probability of this is $P(n) = frac{1}{n}$. Now, the probability of reaching $n+1$ when standing on $n$ is the same as the probability of reaching $0$ when standing on $1$, by symmetry. So we get
$$P(n+1) = P(n)(1 - P(n+1)) Leftrightarrow P(n+1) = frac{P(n)}{1+P(n)} = frac{frac{1}{n}}{frac{n+1}{n}} = frac{1}{n+1}$$
So by the principle of induction, $P(N) = frac{1}{N}$ for all positive integers $N$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1 Nice solution.
    $endgroup$
    – drhab
    Jan 12 at 9:26










  • $begingroup$
    Good answer, but you switches from k to n there
    $endgroup$
    – DreamConspiracy
    Jan 12 at 12:56










  • $begingroup$
    Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
    $endgroup$
    – Pedro A
    Jan 12 at 13:13












  • $begingroup$
    The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
    $endgroup$
    – nesHan
    Jan 12 at 14:01














6












6








6





$begingroup$

Here is a short proof using induction and symmetry. We use induction on $N$ to show that $P(N) = frac{1}{N}$. For $N = 1$, this is obvious.



Now, suppose that $P(k) = frac{1}{k}$ for $k leq n$. We shall show that the statement holds for $N = n+1$. Note that, in order to reach $n+1$, we must reach $n$. The probability of this is $P(n) = frac{1}{n}$. Now, the probability of reaching $n+1$ when standing on $n$ is the same as the probability of reaching $0$ when standing on $1$, by symmetry. So we get
$$P(n+1) = P(n)(1 - P(n+1)) Leftrightarrow P(n+1) = frac{P(n)}{1+P(n)} = frac{frac{1}{n}}{frac{n+1}{n}} = frac{1}{n+1}$$
So by the principle of induction, $P(N) = frac{1}{N}$ for all positive integers $N$






share|cite|improve this answer











$endgroup$



Here is a short proof using induction and symmetry. We use induction on $N$ to show that $P(N) = frac{1}{N}$. For $N = 1$, this is obvious.



Now, suppose that $P(k) = frac{1}{k}$ for $k leq n$. We shall show that the statement holds for $N = n+1$. Note that, in order to reach $n+1$, we must reach $n$. The probability of this is $P(n) = frac{1}{n}$. Now, the probability of reaching $n+1$ when standing on $n$ is the same as the probability of reaching $0$ when standing on $1$, by symmetry. So we get
$$P(n+1) = P(n)(1 - P(n+1)) Leftrightarrow P(n+1) = frac{P(n)}{1+P(n)} = frac{frac{1}{n}}{frac{n+1}{n}} = frac{1}{n+1}$$
So by the principle of induction, $P(N) = frac{1}{N}$ for all positive integers $N$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 12 at 9:26









drhab

101k544130




101k544130










answered Jan 12 at 8:21









nesHannesHan

1735




1735












  • $begingroup$
    +1 Nice solution.
    $endgroup$
    – drhab
    Jan 12 at 9:26










  • $begingroup$
    Good answer, but you switches from k to n there
    $endgroup$
    – DreamConspiracy
    Jan 12 at 12:56










  • $begingroup$
    Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
    $endgroup$
    – Pedro A
    Jan 12 at 13:13












  • $begingroup$
    The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
    $endgroup$
    – nesHan
    Jan 12 at 14:01


















  • $begingroup$
    +1 Nice solution.
    $endgroup$
    – drhab
    Jan 12 at 9:26










  • $begingroup$
    Good answer, but you switches from k to n there
    $endgroup$
    – DreamConspiracy
    Jan 12 at 12:56










  • $begingroup$
    Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
    $endgroup$
    – Pedro A
    Jan 12 at 13:13












  • $begingroup$
    The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
    $endgroup$
    – nesHan
    Jan 12 at 14:01
















$begingroup$
+1 Nice solution.
$endgroup$
– drhab
Jan 12 at 9:26




$begingroup$
+1 Nice solution.
$endgroup$
– drhab
Jan 12 at 9:26












$begingroup$
Good answer, but you switches from k to n there
$endgroup$
– DreamConspiracy
Jan 12 at 12:56




$begingroup$
Good answer, but you switches from k to n there
$endgroup$
– DreamConspiracy
Jan 12 at 12:56












$begingroup$
Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
$endgroup$
– Pedro A
Jan 12 at 13:13






$begingroup$
Why is the probability of reaching 0 standing on 1 equal to $1 - P(n+1)$? Did the "standing on 1" have any relevance to that value?
$endgroup$
– Pedro A
Jan 12 at 13:13














$begingroup$
The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
$endgroup$
– nesHan
Jan 12 at 14:01




$begingroup$
The game ends when we reach either $n+1$ or $0$, and the probability of reaching $n+1$ standing on $1$ is (by definition) $P(n+1)$ so the probability of reaching $0$ becomes $1 - P(n+1)$.
$endgroup$
– nesHan
Jan 12 at 14:01



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