Two players alternate flipping a coin until the result is head. How to derive that the probability for the first player to win is $2/3$? [duplicate]

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5












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This question already has an answer here:



  • Two players alternately flip a coin; what is the probability of winning by getting a head?

    5 answers




Two players, $A$ and $B$, alternately and independently flip a coin and
the first player to obtain a head wins. Player $A$ flips first. What is
the probability that $A$ wins?



Official answer: $2/3$, but I cannot arrive at it.




Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.



When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.



However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.










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marked as duplicate by Key Flex, Theo Bendit, José Carlos Santos, Kemono Chen, mrtaurho Feb 10 at 10:46


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
    $endgroup$
    – Pakk
    Feb 8 at 21:41






  • 1




    $begingroup$
    I have understood the flaw in my method by understanding the correct answer, but thank you anyways
    $endgroup$
    – Victor S.
    Feb 8 at 21:56















5












$begingroup$



This question already has an answer here:



  • Two players alternately flip a coin; what is the probability of winning by getting a head?

    5 answers




Two players, $A$ and $B$, alternately and independently flip a coin and
the first player to obtain a head wins. Player $A$ flips first. What is
the probability that $A$ wins?



Official answer: $2/3$, but I cannot arrive at it.




Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.



When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.



However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.










share|cite|improve this question











$endgroup$



marked as duplicate by Key Flex, Theo Bendit, José Carlos Santos, Kemono Chen, mrtaurho Feb 10 at 10:46


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
    $endgroup$
    – Pakk
    Feb 8 at 21:41






  • 1




    $begingroup$
    I have understood the flaw in my method by understanding the correct answer, but thank you anyways
    $endgroup$
    – Victor S.
    Feb 8 at 21:56













5












5








5





$begingroup$



This question already has an answer here:



  • Two players alternately flip a coin; what is the probability of winning by getting a head?

    5 answers




Two players, $A$ and $B$, alternately and independently flip a coin and
the first player to obtain a head wins. Player $A$ flips first. What is
the probability that $A$ wins?



Official answer: $2/3$, but I cannot arrive at it.




Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.



When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.



However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • Two players alternately flip a coin; what is the probability of winning by getting a head?

    5 answers




Two players, $A$ and $B$, alternately and independently flip a coin and
the first player to obtain a head wins. Player $A$ flips first. What is
the probability that $A$ wins?



Official answer: $2/3$, but I cannot arrive at it.




Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.



When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.



However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.





This question already has an answer here:



  • Two players alternately flip a coin; what is the probability of winning by getting a head?

    5 answers







probability discrete-mathematics binomial-distribution geometric-series






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edited Feb 9 at 8:51









Asaf Karagila

305k33435766




305k33435766










asked Feb 8 at 16:37









Victor S.Victor S.

31019




31019




marked as duplicate by Key Flex, Theo Bendit, José Carlos Santos, Kemono Chen, mrtaurho Feb 10 at 10:46


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Key Flex, Theo Bendit, José Carlos Santos, Kemono Chen, mrtaurho Feb 10 at 10:46


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • $begingroup$
    What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
    $endgroup$
    – Pakk
    Feb 8 at 21:41






  • 1




    $begingroup$
    I have understood the flaw in my method by understanding the correct answer, but thank you anyways
    $endgroup$
    – Victor S.
    Feb 8 at 21:56
















  • $begingroup$
    What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
    $endgroup$
    – Pakk
    Feb 8 at 21:41






  • 1




    $begingroup$
    I have understood the flaw in my method by understanding the correct answer, but thank you anyways
    $endgroup$
    – Victor S.
    Feb 8 at 21:56















$begingroup$
What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
$endgroup$
– Pakk
Feb 8 at 21:41




$begingroup$
What is your question? Do you want to know the answer to the problem, or do you want to know the mistake in your method? If it is the second one, you should make that clear, because all answers so far ignore that.
$endgroup$
– Pakk
Feb 8 at 21:41




1




1




$begingroup$
I have understood the flaw in my method by understanding the correct answer, but thank you anyways
$endgroup$
– Victor S.
Feb 8 at 21:56




$begingroup$
I have understood the flaw in my method by understanding the correct answer, but thank you anyways
$endgroup$
– Victor S.
Feb 8 at 21:56










5 Answers
5






active

oldest

votes


















14












$begingroup$

So $A$ wins if head comes first time in odd number of tosses. Say $P_k$ is probability that head comes first time in $k$-th toss then $P_k = q^k-1p$ and



begineqnarray P &=& P_1+P_3+P_5+... \ \
&=&p+q^2p+q^4p+q^6p+....\ \
&=& pover 1-q^2\ \& =& 1over 1+q
endeqnarray



where $p$ is probability that head comes in one toss and $q=1-p$.



So if the coin is fair, then $p=1/2=q$, so $$P= 2over 3$$






share|cite|improve this answer











$endgroup$




















    23












    $begingroup$

    Let $p$ be the probability that the first player wins. A wins half the time when he tosses a head. The other half the time, B becomes the first tosser, and A wins if the first player doesn't win.$$p=frac12+frac12(1-p)implies p=frac23$$






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Is this way doable even if the coin is not fair?
      $endgroup$
      – greedoid
      Feb 9 at 7:43










    • $begingroup$
      @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
      $endgroup$
      – saulspatz
      Feb 9 at 14:19


















    7












    $begingroup$

    Here's another approach.



    Consider a round of this game as A flipping and then B flipping. Of course the game may end in the middle of a round if A flips heads.



    In the first round A has twice as great a chance to win as B. (A wins half the time by flipping heads); and B wins with tails-heads which occurs one quarter of the time).



    But this two-to-one ratio of A winning to B winning holds in every round. So overall, A is twice as likely to win as B.



    This gives A a $frac23$ chance of winning to $frac13$ for B.






    share|cite|improve this answer









    $endgroup$




















      6












      $begingroup$

      Let $p$ denote the probability that $A$ wins. There is a $.5$ probability $A$ wins on the first flip. If he flips tails, then in order to win $B$ must then flip tails. This situation occurs with probability $.25$ ($1/2 * 1/2$ for both tails) and in this case we are back to where we started. So
      $$p = .5+ .25p$$
      which yields $p=2/3$.






      share|cite|improve this answer









      $endgroup$




















        0












        $begingroup$

        P(A wins) can be thought of a sum of infinite number of cases, starting with >A wins the first round(0.5)




        A loses first round, B loses second round, A wins the third round (0.5)^3




        And so on. This forms an infinite gp with first term 0.5 and common product 0.5^2




        P(A winning) = 0.5/(1-0.25)







        share|cite|improve this answer









        $endgroup$



















          5 Answers
          5






          active

          oldest

          votes








          5 Answers
          5






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          14












          $begingroup$

          So $A$ wins if head comes first time in odd number of tosses. Say $P_k$ is probability that head comes first time in $k$-th toss then $P_k = q^k-1p$ and



          begineqnarray P &=& P_1+P_3+P_5+... \ \
          &=&p+q^2p+q^4p+q^6p+....\ \
          &=& pover 1-q^2\ \& =& 1over 1+q
          endeqnarray



          where $p$ is probability that head comes in one toss and $q=1-p$.



          So if the coin is fair, then $p=1/2=q$, so $$P= 2over 3$$






          share|cite|improve this answer











          $endgroup$

















            14












            $begingroup$

            So $A$ wins if head comes first time in odd number of tosses. Say $P_k$ is probability that head comes first time in $k$-th toss then $P_k = q^k-1p$ and



            begineqnarray P &=& P_1+P_3+P_5+... \ \
            &=&p+q^2p+q^4p+q^6p+....\ \
            &=& pover 1-q^2\ \& =& 1over 1+q
            endeqnarray



            where $p$ is probability that head comes in one toss and $q=1-p$.



            So if the coin is fair, then $p=1/2=q$, so $$P= 2over 3$$






            share|cite|improve this answer











            $endgroup$















              14












              14








              14





              $begingroup$

              So $A$ wins if head comes first time in odd number of tosses. Say $P_k$ is probability that head comes first time in $k$-th toss then $P_k = q^k-1p$ and



              begineqnarray P &=& P_1+P_3+P_5+... \ \
              &=&p+q^2p+q^4p+q^6p+....\ \
              &=& pover 1-q^2\ \& =& 1over 1+q
              endeqnarray



              where $p$ is probability that head comes in one toss and $q=1-p$.



              So if the coin is fair, then $p=1/2=q$, so $$P= 2over 3$$






              share|cite|improve this answer











              $endgroup$



              So $A$ wins if head comes first time in odd number of tosses. Say $P_k$ is probability that head comes first time in $k$-th toss then $P_k = q^k-1p$ and



              begineqnarray P &=& P_1+P_3+P_5+... \ \
              &=&p+q^2p+q^4p+q^6p+....\ \
              &=& pover 1-q^2\ \& =& 1over 1+q
              endeqnarray



              where $p$ is probability that head comes in one toss and $q=1-p$.



              So if the coin is fair, then $p=1/2=q$, so $$P= 2over 3$$







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Feb 9 at 8:27

























              answered Feb 8 at 16:40









              greedoidgreedoid

              45.5k1159114




              45.5k1159114





















                  23












                  $begingroup$

                  Let $p$ be the probability that the first player wins. A wins half the time when he tosses a head. The other half the time, B becomes the first tosser, and A wins if the first player doesn't win.$$p=frac12+frac12(1-p)implies p=frac23$$






                  share|cite|improve this answer









                  $endgroup$












                  • $begingroup$
                    Is this way doable even if the coin is not fair?
                    $endgroup$
                    – greedoid
                    Feb 9 at 7:43










                  • $begingroup$
                    @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                    $endgroup$
                    – saulspatz
                    Feb 9 at 14:19















                  23












                  $begingroup$

                  Let $p$ be the probability that the first player wins. A wins half the time when he tosses a head. The other half the time, B becomes the first tosser, and A wins if the first player doesn't win.$$p=frac12+frac12(1-p)implies p=frac23$$






                  share|cite|improve this answer









                  $endgroup$












                  • $begingroup$
                    Is this way doable even if the coin is not fair?
                    $endgroup$
                    – greedoid
                    Feb 9 at 7:43










                  • $begingroup$
                    @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                    $endgroup$
                    – saulspatz
                    Feb 9 at 14:19













                  23












                  23








                  23





                  $begingroup$

                  Let $p$ be the probability that the first player wins. A wins half the time when he tosses a head. The other half the time, B becomes the first tosser, and A wins if the first player doesn't win.$$p=frac12+frac12(1-p)implies p=frac23$$






                  share|cite|improve this answer









                  $endgroup$



                  Let $p$ be the probability that the first player wins. A wins half the time when he tosses a head. The other half the time, B becomes the first tosser, and A wins if the first player doesn't win.$$p=frac12+frac12(1-p)implies p=frac23$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 8 at 16:43









                  saulspatzsaulspatz

                  16k31331




                  16k31331











                  • $begingroup$
                    Is this way doable even if the coin is not fair?
                    $endgroup$
                    – greedoid
                    Feb 9 at 7:43










                  • $begingroup$
                    @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                    $endgroup$
                    – saulspatz
                    Feb 9 at 14:19
















                  • $begingroup$
                    Is this way doable even if the coin is not fair?
                    $endgroup$
                    – greedoid
                    Feb 9 at 7:43










                  • $begingroup$
                    @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                    $endgroup$
                    – saulspatz
                    Feb 9 at 14:19















                  $begingroup$
                  Is this way doable even if the coin is not fair?
                  $endgroup$
                  – greedoid
                  Feb 9 at 7:43




                  $begingroup$
                  Is this way doable even if the coin is not fair?
                  $endgroup$
                  – greedoid
                  Feb 9 at 7:43












                  $begingroup$
                  @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                  $endgroup$
                  – saulspatz
                  Feb 9 at 14:19




                  $begingroup$
                  @greedoid Sure. If the coin lands heads with probability $h,$ then $p=h+(1-h)(1-p)$
                  $endgroup$
                  – saulspatz
                  Feb 9 at 14:19











                  7












                  $begingroup$

                  Here's another approach.



                  Consider a round of this game as A flipping and then B flipping. Of course the game may end in the middle of a round if A flips heads.



                  In the first round A has twice as great a chance to win as B. (A wins half the time by flipping heads); and B wins with tails-heads which occurs one quarter of the time).



                  But this two-to-one ratio of A winning to B winning holds in every round. So overall, A is twice as likely to win as B.



                  This gives A a $frac23$ chance of winning to $frac13$ for B.






                  share|cite|improve this answer









                  $endgroup$

















                    7












                    $begingroup$

                    Here's another approach.



                    Consider a round of this game as A flipping and then B flipping. Of course the game may end in the middle of a round if A flips heads.



                    In the first round A has twice as great a chance to win as B. (A wins half the time by flipping heads); and B wins with tails-heads which occurs one quarter of the time).



                    But this two-to-one ratio of A winning to B winning holds in every round. So overall, A is twice as likely to win as B.



                    This gives A a $frac23$ chance of winning to $frac13$ for B.






                    share|cite|improve this answer









                    $endgroup$















                      7












                      7








                      7





                      $begingroup$

                      Here's another approach.



                      Consider a round of this game as A flipping and then B flipping. Of course the game may end in the middle of a round if A flips heads.



                      In the first round A has twice as great a chance to win as B. (A wins half the time by flipping heads); and B wins with tails-heads which occurs one quarter of the time).



                      But this two-to-one ratio of A winning to B winning holds in every round. So overall, A is twice as likely to win as B.



                      This gives A a $frac23$ chance of winning to $frac13$ for B.






                      share|cite|improve this answer









                      $endgroup$



                      Here's another approach.



                      Consider a round of this game as A flipping and then B flipping. Of course the game may end in the middle of a round if A flips heads.



                      In the first round A has twice as great a chance to win as B. (A wins half the time by flipping heads); and B wins with tails-heads which occurs one quarter of the time).



                      But this two-to-one ratio of A winning to B winning holds in every round. So overall, A is twice as likely to win as B.



                      This gives A a $frac23$ chance of winning to $frac13$ for B.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Feb 8 at 16:55









                      paw88789paw88789

                      29.3k12349




                      29.3k12349





















                          6












                          $begingroup$

                          Let $p$ denote the probability that $A$ wins. There is a $.5$ probability $A$ wins on the first flip. If he flips tails, then in order to win $B$ must then flip tails. This situation occurs with probability $.25$ ($1/2 * 1/2$ for both tails) and in this case we are back to where we started. So
                          $$p = .5+ .25p$$
                          which yields $p=2/3$.






                          share|cite|improve this answer









                          $endgroup$

















                            6












                            $begingroup$

                            Let $p$ denote the probability that $A$ wins. There is a $.5$ probability $A$ wins on the first flip. If he flips tails, then in order to win $B$ must then flip tails. This situation occurs with probability $.25$ ($1/2 * 1/2$ for both tails) and in this case we are back to where we started. So
                            $$p = .5+ .25p$$
                            which yields $p=2/3$.






                            share|cite|improve this answer









                            $endgroup$















                              6












                              6








                              6





                              $begingroup$

                              Let $p$ denote the probability that $A$ wins. There is a $.5$ probability $A$ wins on the first flip. If he flips tails, then in order to win $B$ must then flip tails. This situation occurs with probability $.25$ ($1/2 * 1/2$ for both tails) and in this case we are back to where we started. So
                              $$p = .5+ .25p$$
                              which yields $p=2/3$.






                              share|cite|improve this answer









                              $endgroup$



                              Let $p$ denote the probability that $A$ wins. There is a $.5$ probability $A$ wins on the first flip. If he flips tails, then in order to win $B$ must then flip tails. This situation occurs with probability $.25$ ($1/2 * 1/2$ for both tails) and in this case we are back to where we started. So
                              $$p = .5+ .25p$$
                              which yields $p=2/3$.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Feb 8 at 16:43









                              pwerthpwerth

                              3,243417




                              3,243417





















                                  0












                                  $begingroup$

                                  P(A wins) can be thought of a sum of infinite number of cases, starting with >A wins the first round(0.5)




                                  A loses first round, B loses second round, A wins the third round (0.5)^3




                                  And so on. This forms an infinite gp with first term 0.5 and common product 0.5^2




                                  P(A winning) = 0.5/(1-0.25)







                                  share|cite|improve this answer









                                  $endgroup$

















                                    0












                                    $begingroup$

                                    P(A wins) can be thought of a sum of infinite number of cases, starting with >A wins the first round(0.5)




                                    A loses first round, B loses second round, A wins the third round (0.5)^3




                                    And so on. This forms an infinite gp with first term 0.5 and common product 0.5^2




                                    P(A winning) = 0.5/(1-0.25)







                                    share|cite|improve this answer









                                    $endgroup$















                                      0












                                      0








                                      0





                                      $begingroup$

                                      P(A wins) can be thought of a sum of infinite number of cases, starting with >A wins the first round(0.5)




                                      A loses first round, B loses second round, A wins the third round (0.5)^3




                                      And so on. This forms an infinite gp with first term 0.5 and common product 0.5^2




                                      P(A winning) = 0.5/(1-0.25)







                                      share|cite|improve this answer









                                      $endgroup$



                                      P(A wins) can be thought of a sum of infinite number of cases, starting with >A wins the first round(0.5)




                                      A loses first round, B loses second round, A wins the third round (0.5)^3




                                      And so on. This forms an infinite gp with first term 0.5 and common product 0.5^2




                                      P(A winning) = 0.5/(1-0.25)








                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Feb 8 at 16:47









                                      Shreyas Shreyas

                                      813




                                      813












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