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 thatHenryHillary 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
asked 4 hours ago
Connor Watson
112
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Check out our Code of Conduct.
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show 3 more comments
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 thatHenryHillary 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
asked 4 hours ago
Connor Watson
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.
Who is "Henry"?
– David G. Stork
3 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
3 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
3 hours ago
2
This is not in fact an expected value problem.
– JMoravitz
3 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
|
show 3 more comments
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 thatHenryHillary 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
asked 4 hours ago
Connor Watson
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.
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 thatHenryHillary 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
probability statistics expected-value
asked 4 hours ago
Connor Watson
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.
asked 4 hours ago
Connor Watson
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.
edited 3 hours ago
asked 4 hours ago
Connor Watson
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.
asked 4 hours ago
Connor Watson
112
asked 4 hours ago
Connor Watson
112
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
3 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
3 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
3 hours ago
2
This is not in fact an expected value problem.
– JMoravitz
3 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
|
show 3 more comments
Who is "Henry"?
– David G. Stork
3 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
3 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
3 hours ago
2
This is not in fact an expected value problem.
– JMoravitz
3 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
3 hours ago
Who is "Henry"?
– David G. Stork
3 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
3 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
3 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
3 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
3 hours ago
2
2
This is not in fact an expected value problem.
– JMoravitz
3 hours ago
This is not in fact an expected value problem.
– JMoravitz
3 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
|
show 3 more comments
2 Answers
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up vote
4
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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 $p^2$
answered 3 hours ago
Ross Millikan
290k23195368
add a comment |
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:
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.
answered 2 hours ago
apnorton
15k33696
add a comment |
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2 Answers
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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 $p^2$
answered 3 hours ago
Ross Millikan
290k23195368
add a comment |
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 $p^2$
answered 3 hours ago
Ross Millikan
290k23195368
add a comment |
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 $p^2$
answered 3 hours ago
Ross Millikan
290k23195368
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 $p^2$
answered 3 hours ago
Ross Millikan
290k23195368
answered 3 hours ago
Ross Millikan
290k23195368
answered 3 hours ago
Ross Millikan
290k23195368
answered 3 hours ago
Ross Millikan
290k23195368
290k23195368
add a comment |
add a comment |
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:
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.
answered 2 hours ago
apnorton
15k33696
add a comment |
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:
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.
answered 2 hours ago
apnorton
15k33696
add a comment |
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:
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.
answered 2 hours ago
apnorton
15k33696
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:
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.
answered 2 hours ago
apnorton
15k33696
answered 2 hours ago
apnorton
15k33696
answered 2 hours ago
apnorton
15k33696
answered 2 hours ago
apnorton
15k33696
15k33696
add a comment |
add a comment |
Connor Watson is a new contributor. Be nice, and check out our Code of Conduct.
Connor Watson is a new contributor. Be nice, and check out our Code of Conduct.
Connor Watson is a new contributor. Be nice, and check out our Code of Conduct.
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Who is "Henry"?
– David G. Stork
3 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
3 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
3 hours ago
2
This is not in fact an expected value problem.
– JMoravitz
3 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