A coin is thrown until the tail is above. Which is the probability of making at most 3 throws.
$begingroup$
A coin is thrown until the tail is above. Which is the probability of making at most 3 throws.
The answer is :$frac{7}{8}$.
I don't understand why. What are the possible cases and what are the favorable ones?
probability
$endgroup$
add a comment |
$begingroup$
A coin is thrown until the tail is above. Which is the probability of making at most 3 throws.
The answer is :$frac{7}{8}$.
I don't understand why. What are the possible cases and what are the favorable ones?
probability
$endgroup$
$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago
add a comment |
$begingroup$
A coin is thrown until the tail is above. Which is the probability of making at most 3 throws.
The answer is :$frac{7}{8}$.
I don't understand why. What are the possible cases and what are the favorable ones?
probability
$endgroup$
A coin is thrown until the tail is above. Which is the probability of making at most 3 throws.
The answer is :$frac{7}{8}$.
I don't understand why. What are the possible cases and what are the favorable ones?
probability
probability
edited 22 mins ago
YuiTo Cheng
2,94741139
2,94741139
asked 4 hours ago
Ica SanduIca Sandu
1499
1499
$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago
add a comment |
$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago
$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago
$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
There are three possibilities: T, HT, HHT with probabilities $1/2$, $1/2cdot1/2$, and $1/2cdot1/2cdot1/2$, resp. They sum up to $7/8$.
$endgroup$
add a comment |
$begingroup$
You will need more than three throws only when head comes out thrice.
Hence
$$p=1-frac1{2^3}.$$
More generally,
$$p=1-frac1{2^n}$$ for the probability of making at most $n$ throws.
$endgroup$
add a comment |
$begingroup$
Let $T$ be the event that a tail is obtained within 3 flips. We want to find $pr(T)$. It is a bit difficult to compute this because we have to consider obtaining tails within 1 flip, 2 flips, AND 3 flips, to which there is overlap. It is a bit easier to compute $pr(T')$ or the probability $T$ does not happen.
For event $T'$ to occur, we would need to not obtain tails within 3 flips, i.e. get heads three times in a row. Since the probability of throwing heads is $frac{1}{2}$, getting three heads in a row is $frac{1}{2^3}=frac{1}{8}$. Hence, $pr(T')=frac{1}{8}$. Since any given event either happens or doesn't, we have in general for event $A$, $$pr(A')=1-pr(A).$$ Therefore,
$$frac{1}{8}=pr(T')=1-pr(T)implies pr(T)=1-frac{1}{8}=frac{7}{8}$$.
$endgroup$
add a comment |
$begingroup$
A really simple way to see this is just to list out all the possible outcomes of flipping a coin three times. There are only $8$ of them:
$$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$
Because the coin flips are independent of each other, each of these combinations are equally likely. So the probability of each is $frac 18$. As you can see, $7$ of them include at least one tail, so the probability of getting a tails in three coin tosses is $frac 78$.
$endgroup$
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
add a comment |
Your Answer
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There are three possibilities: T, HT, HHT with probabilities $1/2$, $1/2cdot1/2$, and $1/2cdot1/2cdot1/2$, resp. They sum up to $7/8$.
$endgroup$
add a comment |
$begingroup$
There are three possibilities: T, HT, HHT with probabilities $1/2$, $1/2cdot1/2$, and $1/2cdot1/2cdot1/2$, resp. They sum up to $7/8$.
$endgroup$
add a comment |
$begingroup$
There are three possibilities: T, HT, HHT with probabilities $1/2$, $1/2cdot1/2$, and $1/2cdot1/2cdot1/2$, resp. They sum up to $7/8$.
$endgroup$
There are three possibilities: T, HT, HHT with probabilities $1/2$, $1/2cdot1/2$, and $1/2cdot1/2cdot1/2$, resp. They sum up to $7/8$.
answered 4 hours ago
Michael HoppeMichael Hoppe
11.3k31837
11.3k31837
add a comment |
add a comment |
$begingroup$
You will need more than three throws only when head comes out thrice.
Hence
$$p=1-frac1{2^3}.$$
More generally,
$$p=1-frac1{2^n}$$ for the probability of making at most $n$ throws.
$endgroup$
add a comment |
$begingroup$
You will need more than three throws only when head comes out thrice.
Hence
$$p=1-frac1{2^3}.$$
More generally,
$$p=1-frac1{2^n}$$ for the probability of making at most $n$ throws.
$endgroup$
add a comment |
$begingroup$
You will need more than three throws only when head comes out thrice.
Hence
$$p=1-frac1{2^3}.$$
More generally,
$$p=1-frac1{2^n}$$ for the probability of making at most $n$ throws.
$endgroup$
You will need more than three throws only when head comes out thrice.
Hence
$$p=1-frac1{2^3}.$$
More generally,
$$p=1-frac1{2^n}$$ for the probability of making at most $n$ throws.
edited 4 hours ago
answered 4 hours ago
Yves DaoustYves Daoust
134k676232
134k676232
add a comment |
add a comment |
$begingroup$
Let $T$ be the event that a tail is obtained within 3 flips. We want to find $pr(T)$. It is a bit difficult to compute this because we have to consider obtaining tails within 1 flip, 2 flips, AND 3 flips, to which there is overlap. It is a bit easier to compute $pr(T')$ or the probability $T$ does not happen.
For event $T'$ to occur, we would need to not obtain tails within 3 flips, i.e. get heads three times in a row. Since the probability of throwing heads is $frac{1}{2}$, getting three heads in a row is $frac{1}{2^3}=frac{1}{8}$. Hence, $pr(T')=frac{1}{8}$. Since any given event either happens or doesn't, we have in general for event $A$, $$pr(A')=1-pr(A).$$ Therefore,
$$frac{1}{8}=pr(T')=1-pr(T)implies pr(T)=1-frac{1}{8}=frac{7}{8}$$.
$endgroup$
add a comment |
$begingroup$
Let $T$ be the event that a tail is obtained within 3 flips. We want to find $pr(T)$. It is a bit difficult to compute this because we have to consider obtaining tails within 1 flip, 2 flips, AND 3 flips, to which there is overlap. It is a bit easier to compute $pr(T')$ or the probability $T$ does not happen.
For event $T'$ to occur, we would need to not obtain tails within 3 flips, i.e. get heads three times in a row. Since the probability of throwing heads is $frac{1}{2}$, getting three heads in a row is $frac{1}{2^3}=frac{1}{8}$. Hence, $pr(T')=frac{1}{8}$. Since any given event either happens or doesn't, we have in general for event $A$, $$pr(A')=1-pr(A).$$ Therefore,
$$frac{1}{8}=pr(T')=1-pr(T)implies pr(T)=1-frac{1}{8}=frac{7}{8}$$.
$endgroup$
add a comment |
$begingroup$
Let $T$ be the event that a tail is obtained within 3 flips. We want to find $pr(T)$. It is a bit difficult to compute this because we have to consider obtaining tails within 1 flip, 2 flips, AND 3 flips, to which there is overlap. It is a bit easier to compute $pr(T')$ or the probability $T$ does not happen.
For event $T'$ to occur, we would need to not obtain tails within 3 flips, i.e. get heads three times in a row. Since the probability of throwing heads is $frac{1}{2}$, getting three heads in a row is $frac{1}{2^3}=frac{1}{8}$. Hence, $pr(T')=frac{1}{8}$. Since any given event either happens or doesn't, we have in general for event $A$, $$pr(A')=1-pr(A).$$ Therefore,
$$frac{1}{8}=pr(T')=1-pr(T)implies pr(T)=1-frac{1}{8}=frac{7}{8}$$.
$endgroup$
Let $T$ be the event that a tail is obtained within 3 flips. We want to find $pr(T)$. It is a bit difficult to compute this because we have to consider obtaining tails within 1 flip, 2 flips, AND 3 flips, to which there is overlap. It is a bit easier to compute $pr(T')$ or the probability $T$ does not happen.
For event $T'$ to occur, we would need to not obtain tails within 3 flips, i.e. get heads three times in a row. Since the probability of throwing heads is $frac{1}{2}$, getting three heads in a row is $frac{1}{2^3}=frac{1}{8}$. Hence, $pr(T')=frac{1}{8}$. Since any given event either happens or doesn't, we have in general for event $A$, $$pr(A')=1-pr(A).$$ Therefore,
$$frac{1}{8}=pr(T')=1-pr(T)implies pr(T)=1-frac{1}{8}=frac{7}{8}$$.
answered 4 hours ago
Jane CooperJane Cooper
536
536
add a comment |
add a comment |
$begingroup$
A really simple way to see this is just to list out all the possible outcomes of flipping a coin three times. There are only $8$ of them:
$$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$
Because the coin flips are independent of each other, each of these combinations are equally likely. So the probability of each is $frac 18$. As you can see, $7$ of them include at least one tail, so the probability of getting a tails in three coin tosses is $frac 78$.
$endgroup$
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
add a comment |
$begingroup$
A really simple way to see this is just to list out all the possible outcomes of flipping a coin three times. There are only $8$ of them:
$$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$
Because the coin flips are independent of each other, each of these combinations are equally likely. So the probability of each is $frac 18$. As you can see, $7$ of them include at least one tail, so the probability of getting a tails in three coin tosses is $frac 78$.
$endgroup$
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
add a comment |
$begingroup$
A really simple way to see this is just to list out all the possible outcomes of flipping a coin three times. There are only $8$ of them:
$$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$
Because the coin flips are independent of each other, each of these combinations are equally likely. So the probability of each is $frac 18$. As you can see, $7$ of them include at least one tail, so the probability of getting a tails in three coin tosses is $frac 78$.
$endgroup$
A really simple way to see this is just to list out all the possible outcomes of flipping a coin three times. There are only $8$ of them:
$$ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$
Because the coin flips are independent of each other, each of these combinations are equally likely. So the probability of each is $frac 18$. As you can see, $7$ of them include at least one tail, so the probability of getting a tails in three coin tosses is $frac 78$.
answered 44 mins ago
Paul SinclairPaul Sinclair
21.1k21543
21.1k21543
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
add a comment |
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
ok but I flipping until the tail is above not 3 times. Then one possible case is, for example HHHHHT or T or HT...
$endgroup$
– Ica Sandu
41 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
That doesn't matter. What you do after finding your tail isn't going to change the probability that you found it in the first place. So, even though there is no particular reason for you to keep flipping after you get a tail, pretend that you do so anyway. You get this scenario, and find that your probability was $7/8$. That holds even though some of the sequences contain unnecessary flips.
$endgroup$
– Paul Sinclair
36 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
As for 3 flips, that is because your question is about 3 flips. It also doesn't matter to that question how many more times you flip the coin after those three flips are over. Their outcome is already decided.
$endgroup$
– Paul Sinclair
34 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
$begingroup$
I will think a time. I must undestand ,and i will accept your answer.
$endgroup$
– Ica Sandu
22 mins ago
add a comment |
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$begingroup$
It may be useful to know that this is a geometrically distributed random variable.
$endgroup$
– Peter Foreman
4 hours ago