a coin tossing problem
$begingroup$
A and B are playing coin tossing games. If there is a 2 continuous heads, then A win. If there is a 3 continuous tails, then B win. They keep tossing the coin until one of them win. what's the probability of A wins the game?
probability
$endgroup$
add a comment |
$begingroup$
A and B are playing coin tossing games. If there is a 2 continuous heads, then A win. If there is a 3 continuous tails, then B win. They keep tossing the coin until one of them win. what's the probability of A wins the game?
probability
$endgroup$
add a comment |
$begingroup$
A and B are playing coin tossing games. If there is a 2 continuous heads, then A win. If there is a 3 continuous tails, then B win. They keep tossing the coin until one of them win. what's the probability of A wins the game?
probability
$endgroup$
A and B are playing coin tossing games. If there is a 2 continuous heads, then A win. If there is a 3 continuous tails, then B win. They keep tossing the coin until one of them win. what's the probability of A wins the game?
probability
probability
asked Dec 7 '18 at 5:35
Mango SickMango Sick
91
91
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint:
For A to win, it needs to get $2H$ before B gets $3T$.
So, possible cass may be: $$(H,H),(T,H,H), (T,T,H,H)$$
$$P(HH)=frac{1}{2^2}$$
$$P(THH)=frac{1}{2^3}$$
$$P(TTHH)=frac{1}{2^4}$$
There can be more possibilities such as $$(H,T,T,H,H),(T,T,H,T,H,H) text{ etc. }$$
Try to generalize for $n$ tosses, and then solve for $nto infty$
$endgroup$
add a comment |
$begingroup$
The probability vector $[mathbb{P}(text{A wins}), mathbb{P}(text{B wins})]$ is given by the average of the first and third row of (matrices typed like when inputted to for example to WolframAlpha)
{ inverse (1/2*[[2,-1,-1,0,0], [0,2,-1,0,0],[-1,0,2,-1,0], [-1,0,0,2,-1], [-1,0,0,0,2]] ) } [[0,0], [1/2,0], [0,0], [0,0], [0,1/2]]
and is $[frac{15}{22}, frac{7}{22}]$.
In this solution we are calculating the absorption probabilities of a Markov chain with states $H, HH, T, TT, TTT, A, B$ (with obvious meanings and transition probabilities).
Here' how the chain looks like (all transitions have probability $frac{1}{2}$ except the absorbing winning states $A$ and $B$ of course have self-transition with probability $1$) I included the starting state $S$ here but not in the calculations where this is accounted for by averaging whether you start from $H$ or $T$ (whether the first toss is heads or tails):
$endgroup$
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
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%2f3029517%2fa-coin-tossing-problem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
For A to win, it needs to get $2H$ before B gets $3T$.
So, possible cass may be: $$(H,H),(T,H,H), (T,T,H,H)$$
$$P(HH)=frac{1}{2^2}$$
$$P(THH)=frac{1}{2^3}$$
$$P(TTHH)=frac{1}{2^4}$$
There can be more possibilities such as $$(H,T,T,H,H),(T,T,H,T,H,H) text{ etc. }$$
Try to generalize for $n$ tosses, and then solve for $nto infty$
$endgroup$
add a comment |
$begingroup$
Hint:
For A to win, it needs to get $2H$ before B gets $3T$.
So, possible cass may be: $$(H,H),(T,H,H), (T,T,H,H)$$
$$P(HH)=frac{1}{2^2}$$
$$P(THH)=frac{1}{2^3}$$
$$P(TTHH)=frac{1}{2^4}$$
There can be more possibilities such as $$(H,T,T,H,H),(T,T,H,T,H,H) text{ etc. }$$
Try to generalize for $n$ tosses, and then solve for $nto infty$
$endgroup$
add a comment |
$begingroup$
Hint:
For A to win, it needs to get $2H$ before B gets $3T$.
So, possible cass may be: $$(H,H),(T,H,H), (T,T,H,H)$$
$$P(HH)=frac{1}{2^2}$$
$$P(THH)=frac{1}{2^3}$$
$$P(TTHH)=frac{1}{2^4}$$
There can be more possibilities such as $$(H,T,T,H,H),(T,T,H,T,H,H) text{ etc. }$$
Try to generalize for $n$ tosses, and then solve for $nto infty$
$endgroup$
Hint:
For A to win, it needs to get $2H$ before B gets $3T$.
So, possible cass may be: $$(H,H),(T,H,H), (T,T,H,H)$$
$$P(HH)=frac{1}{2^2}$$
$$P(THH)=frac{1}{2^3}$$
$$P(TTHH)=frac{1}{2^4}$$
There can be more possibilities such as $$(H,T,T,H,H),(T,T,H,T,H,H) text{ etc. }$$
Try to generalize for $n$ tosses, and then solve for $nto infty$
edited Dec 7 '18 at 5:59
answered Dec 7 '18 at 5:53
ideaidea
2,15841125
2,15841125
add a comment |
add a comment |
$begingroup$
The probability vector $[mathbb{P}(text{A wins}), mathbb{P}(text{B wins})]$ is given by the average of the first and third row of (matrices typed like when inputted to for example to WolframAlpha)
{ inverse (1/2*[[2,-1,-1,0,0], [0,2,-1,0,0],[-1,0,2,-1,0], [-1,0,0,2,-1], [-1,0,0,0,2]] ) } [[0,0], [1/2,0], [0,0], [0,0], [0,1/2]]
and is $[frac{15}{22}, frac{7}{22}]$.
In this solution we are calculating the absorption probabilities of a Markov chain with states $H, HH, T, TT, TTT, A, B$ (with obvious meanings and transition probabilities).
Here' how the chain looks like (all transitions have probability $frac{1}{2}$ except the absorbing winning states $A$ and $B$ of course have self-transition with probability $1$) I included the starting state $S$ here but not in the calculations where this is accounted for by averaging whether you start from $H$ or $T$ (whether the first toss is heads or tails):
$endgroup$
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
add a comment |
$begingroup$
The probability vector $[mathbb{P}(text{A wins}), mathbb{P}(text{B wins})]$ is given by the average of the first and third row of (matrices typed like when inputted to for example to WolframAlpha)
{ inverse (1/2*[[2,-1,-1,0,0], [0,2,-1,0,0],[-1,0,2,-1,0], [-1,0,0,2,-1], [-1,0,0,0,2]] ) } [[0,0], [1/2,0], [0,0], [0,0], [0,1/2]]
and is $[frac{15}{22}, frac{7}{22}]$.
In this solution we are calculating the absorption probabilities of a Markov chain with states $H, HH, T, TT, TTT, A, B$ (with obvious meanings and transition probabilities).
Here' how the chain looks like (all transitions have probability $frac{1}{2}$ except the absorbing winning states $A$ and $B$ of course have self-transition with probability $1$) I included the starting state $S$ here but not in the calculations where this is accounted for by averaging whether you start from $H$ or $T$ (whether the first toss is heads or tails):
$endgroup$
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
add a comment |
$begingroup$
The probability vector $[mathbb{P}(text{A wins}), mathbb{P}(text{B wins})]$ is given by the average of the first and third row of (matrices typed like when inputted to for example to WolframAlpha)
{ inverse (1/2*[[2,-1,-1,0,0], [0,2,-1,0,0],[-1,0,2,-1,0], [-1,0,0,2,-1], [-1,0,0,0,2]] ) } [[0,0], [1/2,0], [0,0], [0,0], [0,1/2]]
and is $[frac{15}{22}, frac{7}{22}]$.
In this solution we are calculating the absorption probabilities of a Markov chain with states $H, HH, T, TT, TTT, A, B$ (with obvious meanings and transition probabilities).
Here' how the chain looks like (all transitions have probability $frac{1}{2}$ except the absorbing winning states $A$ and $B$ of course have self-transition with probability $1$) I included the starting state $S$ here but not in the calculations where this is accounted for by averaging whether you start from $H$ or $T$ (whether the first toss is heads or tails):
$endgroup$
The probability vector $[mathbb{P}(text{A wins}), mathbb{P}(text{B wins})]$ is given by the average of the first and third row of (matrices typed like when inputted to for example to WolframAlpha)
{ inverse (1/2*[[2,-1,-1,0,0], [0,2,-1,0,0],[-1,0,2,-1,0], [-1,0,0,2,-1], [-1,0,0,0,2]] ) } [[0,0], [1/2,0], [0,0], [0,0], [0,1/2]]
and is $[frac{15}{22}, frac{7}{22}]$.
In this solution we are calculating the absorption probabilities of a Markov chain with states $H, HH, T, TT, TTT, A, B$ (with obvious meanings and transition probabilities).
Here' how the chain looks like (all transitions have probability $frac{1}{2}$ except the absorbing winning states $A$ and $B$ of course have self-transition with probability $1$) I included the starting state $S$ here but not in the calculations where this is accounted for by averaging whether you start from $H$ or $T$ (whether the first toss is heads or tails):
edited Dec 8 '18 at 10:32
answered Dec 7 '18 at 12:09
ploosu2ploosu2
4,6451024
4,6451024
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
add a comment |
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
$begingroup$
For some reason, WolframAlpha wouldn't show the result correctly if you put the last $frac{1}{2}$ inside the matrix like I have done above. Take it out and remember to divide by $2$ yourself if this also happens to you, i.e insert [[0,0], [1,0], [0,0], [0,0], [0,1]] instead in the above formula.
$endgroup$
– ploosu2
Dec 7 '18 at 12:12
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%2f3029517%2fa-coin-tossing-problem%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