Mixing
Price of a n toss coin game
$begingroup$
I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their 1 dollar bet plus an additional 1 dollar. If the outcome is a tails they lose their 1 dollar bet and get nothing.
If n = 1 the fair price will be 0.5 dollars to play. But what happens if we want to make it n tosses? It becomes complicated because at each iterative step the player has to decide whether the current pay off is greater or less then the expected outcome of tossing another coin.
I simulated this once and got a fair price to be 0.76 dollars for n = 20 but I just cannot recall what my logic was.
Any suggestions will be appreciated.
The players pay off is the number of heads in n tosses.
probability statistics
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
$endgroup$
|
show 2 more comments
$begingroup$
I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their 1 dollar bet plus an additional 1 dollar. If the outcome is a tails they lose their 1 dollar bet and get nothing.
If n = 1 the fair price will be 0.5 dollars to play. But what happens if we want to make it n tosses? It becomes complicated because at each iterative step the player has to decide whether the current pay off is greater or less then the expected outcome of tossing another coin.
I simulated this once and got a fair price to be 0.76 dollars for n = 20 but I just cannot recall what my logic was.
Any suggestions will be appreciated.
The players pay off is the number of heads in n tosses.
probability statistics
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
$endgroup$
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54
|
show 2 more comments
$begingroup$
I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their 1 dollar bet plus an additional 1 dollar. If the outcome is a tails they lose their 1 dollar bet and get nothing.
If n = 1 the fair price will be 0.5 dollars to play. But what happens if we want to make it n tosses? It becomes complicated because at each iterative step the player has to decide whether the current pay off is greater or less then the expected outcome of tossing another coin.
I simulated this once and got a fair price to be 0.76 dollars for n = 20 but I just cannot recall what my logic was.
Any suggestions will be appreciated.
The players pay off is the number of heads in n tosses.
probability statistics
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
$endgroup$
I was thinking of a game where a player pays one dollar to toss a coin. If the result is a head they get their 1 dollar bet plus an additional 1 dollar. If the outcome is a tails they lose their 1 dollar bet and get nothing.
If n = 1 the fair price will be 0.5 dollars to play. But what happens if we want to make it n tosses? It becomes complicated because at each iterative step the player has to decide whether the current pay off is greater or less then the expected outcome of tossing another coin.
I simulated this once and got a fair price to be 0.76 dollars for n = 20 but I just cannot recall what my logic was.
Any suggestions will be appreciated.
The players pay off is the number of heads in n tosses.
probability statistics
probability statistics
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
asked Feb 15 '14 at 7:27
Seeking AlphaSeeking Alpha
25228
25228
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54
|
show 2 more comments
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The fair price for $n$ tosses is $n$ dollars.
JiK has shown for a single toss the fair price is one dollar. For $n$ tosses if the player pays a dollar for each toss then the expected profit is $$E[mathrm{gain}]= {left( { n over 2} right)} (+1) + {left( { n over 2} right) } (-1)= 0,$$ where ${n over 2}$ is the expected number of heads and the expected number of tails. So the fair price is $n.$
Intuitively, this makes sense. All tosses are independent, so if the fair price for a single toss is one dollar, then the fair price for $n$ tosses should be $n$ dollars.
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f677147%2fprice-of-a-n-toss-coin-game%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The fair price for $n$ tosses is $n$ dollars.
JiK has shown for a single toss the fair price is one dollar. For $n$ tosses if the player pays a dollar for each toss then the expected profit is $$E[mathrm{gain}]= {left( { n over 2} right)} (+1) + {left( { n over 2} right) } (-1)= 0,$$ where ${n over 2}$ is the expected number of heads and the expected number of tails. So the fair price is $n.$
Intuitively, this makes sense. All tosses are independent, so if the fair price for a single toss is one dollar, then the fair price for $n$ tosses should be $n$ dollars.
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
$endgroup$
add a comment |
$begingroup$
The fair price for $n$ tosses is $n$ dollars.
JiK has shown for a single toss the fair price is one dollar. For $n$ tosses if the player pays a dollar for each toss then the expected profit is $$E[mathrm{gain}]= {left( { n over 2} right)} (+1) + {left( { n over 2} right) } (-1)= 0,$$ where ${n over 2}$ is the expected number of heads and the expected number of tails. So the fair price is $n.$
Intuitively, this makes sense. All tosses are independent, so if the fair price for a single toss is one dollar, then the fair price for $n$ tosses should be $n$ dollars.
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
$endgroup$
add a comment |
$begingroup$
The fair price for $n$ tosses is $n$ dollars.
JiK has shown for a single toss the fair price is one dollar. For $n$ tosses if the player pays a dollar for each toss then the expected profit is $$E[mathrm{gain}]= {left( { n over 2} right)} (+1) + {left( { n over 2} right) } (-1)= 0,$$ where ${n over 2}$ is the expected number of heads and the expected number of tails. So the fair price is $n.$
Intuitively, this makes sense. All tosses are independent, so if the fair price for a single toss is one dollar, then the fair price for $n$ tosses should be $n$ dollars.
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
$endgroup$
The fair price for $n$ tosses is $n$ dollars.
JiK has shown for a single toss the fair price is one dollar. For $n$ tosses if the player pays a dollar for each toss then the expected profit is $$E[mathrm{gain}]= {left( { n over 2} right)} (+1) + {left( { n over 2} right) } (-1)= 0,$$ where ${n over 2}$ is the expected number of heads and the expected number of tails. So the fair price is $n.$
Intuitively, this makes sense. All tosses are independent, so if the fair price for a single toss is one dollar, then the fair price for $n$ tosses should be $n$ dollars.
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
answered Feb 16 '14 at 3:09
soakleysoakley
1,42621114
1,42621114
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f677147%2fprice-of-a-n-toss-coin-game%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
define "fair price" in "fair price will be 0.5"
$endgroup$
– Roam
Feb 15 '14 at 7:36
$begingroup$
if you were going to sell this opportunity to a unlimited number of punters if you charge 50c for each person to play in the long run you will break even ie. you would expect not to make money and not to lose money
$endgroup$
– Seeking Alpha
Feb 15 '14 at 7:38
$begingroup$
@SeekingAlpha How did you get 50c? If $n=1$, you lose one dollar with probability $frac{1}{2}$ and win one dollar with probability $frac{1}{2}$.
$endgroup$
– JiK
Feb 15 '14 at 8:48
$begingroup$
@JiK, $frac 12 cdot 1+frac 12cdot0=$1=50 c$.
$endgroup$
– Ragnar
Feb 15 '14 at 8:52
$begingroup$
What do you earn after two tosses? Does one earn $1$ per head or only when all tosses are head?
$endgroup$
– Ragnar
Feb 15 '14 at 8:54