Probability of Winning Coin Tosses - Variable Number of Games











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Hillary and Trump play a game of coin toss. The coin is fair such that $mathrm{Pr}(x=H) = mathrm{Pr}(x=T) = 0.5$. If it gets a Head (H), Hillary wins, otherwise, Trump wins.
They agree in advance that the first player who has won $3$ rounds will collect the entire prize. Coin flipping, however, is interrupted for some reason after $3$ rounds and they got $1$ Head and $2$ Tails. Suppose that they continue to toss the coin afterwards, what is the probability that Henry Hillary will win the entire prize?




I saw this question also on Chegg and I got the correct answer, but I drew a tree of the rest of the remaining outcomes. We get: 3T, 3HT, and 3HH. Hillary winning would be HH, so $mathrm{Pr}(HH) = 3/9 = 1/3$. But I want to know the proper way to do this problem. For example, what formula is used?



The second the coin changed to be biased, I have no idea what to do. In addition, how would you derive the solution is the coin is biased? Ex: Pr(x=H)=0.75? I believe this is an Expected Value problem...looked through some other questions and I couldn't quite figure out the solution.



Any solutions or links to duplicate questions (with solutions) would be great. Thank you in advance!



P.S. I saw this question but it didn't quite help me...
Fair and Unfair coin Probability










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  • Who is "Henry"?
    – David G. Stork
    4 hours ago










  • You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
    – JMoravitz
    4 hours ago










  • Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
    – Connor Watson
    4 hours ago






  • 2




    This is not in fact an expected value problem.
    – JMoravitz
    4 hours ago






  • 1




    Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
    – Connor Watson
    3 hours ago















up vote
2
down vote

favorite













Hillary and Trump play a game of coin toss. The coin is fair such that $mathrm{Pr}(x=H) = mathrm{Pr}(x=T) = 0.5$. If it gets a Head (H), Hillary wins, otherwise, Trump wins.
They agree in advance that the first player who has won $3$ rounds will collect the entire prize. Coin flipping, however, is interrupted for some reason after $3$ rounds and they got $1$ Head and $2$ Tails. Suppose that they continue to toss the coin afterwards, what is the probability that Henry Hillary will win the entire prize?




I saw this question also on Chegg and I got the correct answer, but I drew a tree of the rest of the remaining outcomes. We get: 3T, 3HT, and 3HH. Hillary winning would be HH, so $mathrm{Pr}(HH) = 3/9 = 1/3$. But I want to know the proper way to do this problem. For example, what formula is used?



The second the coin changed to be biased, I have no idea what to do. In addition, how would you derive the solution is the coin is biased? Ex: Pr(x=H)=0.75? I believe this is an Expected Value problem...looked through some other questions and I couldn't quite figure out the solution.



Any solutions or links to duplicate questions (with solutions) would be great. Thank you in advance!



P.S. I saw this question but it didn't quite help me...
Fair and Unfair coin Probability










share|cite|improve this question









New contributor




Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Who is "Henry"?
    – David G. Stork
    4 hours ago










  • You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
    – JMoravitz
    4 hours ago










  • Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
    – Connor Watson
    4 hours ago






  • 2




    This is not in fact an expected value problem.
    – JMoravitz
    4 hours ago






  • 1




    Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
    – Connor Watson
    3 hours ago













up vote
2
down vote

favorite









up vote
2
down vote

favorite












Hillary and Trump play a game of coin toss. The coin is fair such that $mathrm{Pr}(x=H) = mathrm{Pr}(x=T) = 0.5$. If it gets a Head (H), Hillary wins, otherwise, Trump wins.
They agree in advance that the first player who has won $3$ rounds will collect the entire prize. Coin flipping, however, is interrupted for some reason after $3$ rounds and they got $1$ Head and $2$ Tails. Suppose that they continue to toss the coin afterwards, what is the probability that Henry Hillary will win the entire prize?




I saw this question also on Chegg and I got the correct answer, but I drew a tree of the rest of the remaining outcomes. We get: 3T, 3HT, and 3HH. Hillary winning would be HH, so $mathrm{Pr}(HH) = 3/9 = 1/3$. But I want to know the proper way to do this problem. For example, what formula is used?



The second the coin changed to be biased, I have no idea what to do. In addition, how would you derive the solution is the coin is biased? Ex: Pr(x=H)=0.75? I believe this is an Expected Value problem...looked through some other questions and I couldn't quite figure out the solution.



Any solutions or links to duplicate questions (with solutions) would be great. Thank you in advance!



P.S. I saw this question but it didn't quite help me...
Fair and Unfair coin Probability










share|cite|improve this question









New contributor




Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












Hillary and Trump play a game of coin toss. The coin is fair such that $mathrm{Pr}(x=H) = mathrm{Pr}(x=T) = 0.5$. If it gets a Head (H), Hillary wins, otherwise, Trump wins.
They agree in advance that the first player who has won $3$ rounds will collect the entire prize. Coin flipping, however, is interrupted for some reason after $3$ rounds and they got $1$ Head and $2$ Tails. Suppose that they continue to toss the coin afterwards, what is the probability that Henry Hillary will win the entire prize?




I saw this question also on Chegg and I got the correct answer, but I drew a tree of the rest of the remaining outcomes. We get: 3T, 3HT, and 3HH. Hillary winning would be HH, so $mathrm{Pr}(HH) = 3/9 = 1/3$. But I want to know the proper way to do this problem. For example, what formula is used?



The second the coin changed to be biased, I have no idea what to do. In addition, how would you derive the solution is the coin is biased? Ex: Pr(x=H)=0.75? I believe this is an Expected Value problem...looked through some other questions and I couldn't quite figure out the solution.



Any solutions or links to duplicate questions (with solutions) would be great. Thank you in advance!



P.S. I saw this question but it didn't quite help me...
Fair and Unfair coin Probability







probability statistics expected-value






share|cite|improve this question









New contributor




Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




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edited 3 hours ago





















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Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 4 hours ago









Connor Watson

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112




New contributor




Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Connor Watson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Who is "Henry"?
    – David G. Stork
    4 hours ago










  • You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
    – JMoravitz
    4 hours ago










  • Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
    – Connor Watson
    4 hours ago






  • 2




    This is not in fact an expected value problem.
    – JMoravitz
    4 hours ago






  • 1




    Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
    – Connor Watson
    3 hours ago


















  • Who is "Henry"?
    – David G. Stork
    4 hours ago










  • You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
    – JMoravitz
    4 hours ago










  • Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
    – Connor Watson
    4 hours ago






  • 2




    This is not in fact an expected value problem.
    – JMoravitz
    4 hours ago






  • 1




    Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
    – Connor Watson
    3 hours ago
















Who is "Henry"?
– David G. Stork
4 hours ago




Who is "Henry"?
– David G. Stork
4 hours ago












You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
– JMoravitz
4 hours ago




You can continue tossing coins even after an overall winner is determined. By the fifth coin-flip there is going to be exactly one person with at least 3 wins and that person is the overall winner. Use binomial distribution to calculate the probabilities of a player having won exactly 3, exactly 4, or exactly 5 of the flips over the course of the series and add those probabilities together. If told that certain outcomes have already happened, then only do a binomial distribution over the still remaining games to play until 5 happen.
– JMoravitz
4 hours ago












Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
– Connor Watson
4 hours ago




Thank you for highlighting the actual question, and for pointing out the typo. Henry is supposed to be Hillary (the original question has Henry).
– Connor Watson
4 hours ago




2




2




This is not in fact an expected value problem.
– JMoravitz
4 hours ago




This is not in fact an expected value problem.
– JMoravitz
4 hours ago




1




1




Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
– Connor Watson
3 hours ago




Hi all, I made a clarification in the question. But, I am not looking for the probability that any player wins. I'm looking for the probability that Hillary wins, which actually isn't determined by an exact number of coin flips. As shown, the first player to win 3 rounds wins, which means the game can end on the next flip (if trump gets a tails), being a total of either 4 or 5 rounds. The probability shifts when the coin is biased.
– Connor Watson
3 hours ago










2 Answers
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Hillary wins if the next two flips are $HH$, which is probability $frac 14$. Otherwise Trump wins. If the coin is not fair and falls heads with probability $p$ her chance of winning is $p^2$






share|cite|improve this answer




























    up vote
    2
    down vote













    Drawing a tree (or even a partial tree) is the way I'd recommend approaching this sort of problem, at least if you're dealing with a reasonably-sized case set:



    Tree of outcomes



    If you're dealing with an unfair coin, I find it helps to draw the probabilities on the edges. Where we start, you can see that we flip the coin and get heads with probability $p$. To get heads again (and for Hillary/Henry) to win, we have to get heads again, so we multiply by $p$ (independent events). This gives us a probability of $p^2$ to get two heads and for Hillary to win.






    share|cite|improve this answer





















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      2 Answers
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      up vote
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      Hillary wins if the next two flips are $HH$, which is probability $frac 14$. Otherwise Trump wins. If the coin is not fair and falls heads with probability $p$ her chance of winning is $p^2$






      share|cite|improve this answer

























        up vote
        4
        down vote













        Hillary wins if the next two flips are $HH$, which is probability $frac 14$. Otherwise Trump wins. If the coin is not fair and falls heads with probability $p$ her chance of winning is $p^2$






        share|cite|improve this answer























          up vote
          4
          down vote










          up vote
          4
          down vote









          Hillary wins if the next two flips are $HH$, which is probability $frac 14$. Otherwise Trump wins. If the coin is not fair and falls heads with probability $p$ her chance of winning is $p^2$






          share|cite|improve this answer












          Hillary wins if the next two flips are $HH$, which is probability $frac 14$. Otherwise Trump wins. If the coin is not fair and falls heads with probability $p$ her chance of winning is $p^2$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Ross Millikan

          290k23195368




          290k23195368






















              up vote
              2
              down vote













              Drawing a tree (or even a partial tree) is the way I'd recommend approaching this sort of problem, at least if you're dealing with a reasonably-sized case set:



              Tree of outcomes



              If you're dealing with an unfair coin, I find it helps to draw the probabilities on the edges. Where we start, you can see that we flip the coin and get heads with probability $p$. To get heads again (and for Hillary/Henry) to win, we have to get heads again, so we multiply by $p$ (independent events). This gives us a probability of $p^2$ to get two heads and for Hillary to win.






              share|cite|improve this answer

























                up vote
                2
                down vote













                Drawing a tree (or even a partial tree) is the way I'd recommend approaching this sort of problem, at least if you're dealing with a reasonably-sized case set:



                Tree of outcomes



                If you're dealing with an unfair coin, I find it helps to draw the probabilities on the edges. Where we start, you can see that we flip the coin and get heads with probability $p$. To get heads again (and for Hillary/Henry) to win, we have to get heads again, so we multiply by $p$ (independent events). This gives us a probability of $p^2$ to get two heads and for Hillary to win.






                share|cite|improve this answer























                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  Drawing a tree (or even a partial tree) is the way I'd recommend approaching this sort of problem, at least if you're dealing with a reasonably-sized case set:



                  Tree of outcomes



                  If you're dealing with an unfair coin, I find it helps to draw the probabilities on the edges. Where we start, you can see that we flip the coin and get heads with probability $p$. To get heads again (and for Hillary/Henry) to win, we have to get heads again, so we multiply by $p$ (independent events). This gives us a probability of $p^2$ to get two heads and for Hillary to win.






                  share|cite|improve this answer












                  Drawing a tree (or even a partial tree) is the way I'd recommend approaching this sort of problem, at least if you're dealing with a reasonably-sized case set:



                  Tree of outcomes



                  If you're dealing with an unfair coin, I find it helps to draw the probabilities on the edges. Where we start, you can see that we flip the coin and get heads with probability $p$. To get heads again (and for Hillary/Henry) to win, we have to get heads again, so we multiply by $p$ (independent events). This gives us a probability of $p^2$ to get two heads and for Hillary to win.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  apnorton

                  15k33696




                  15k33696






















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