Probability of getting 6 heads in a row from 200 flips and intuition about this high value
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A few days ago i had an argument with a friend about this question :
What is the probability of getting 6 heads in a row from 200 flips ?
I argued it is high probability (significantly bigger than half) while he argued it is low probability.
When i tried to give exact formula i failed so we checked the web were the answer was about 84%, yet he is still not convinced so from this i have two questions:
1) What is the exact formula for $k$ Heads in a row (consecutive) out of $n$ coin flips?
2) (Not a mathematical) How to convince my friend that 6 in a row have high probability ? meaning what is the intuition behind the question ?
probability intuition
edited Dec 29 '18 at 10:58
Did
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
$endgroup$
|
show 2 more comments
$begingroup$
A few days ago i had an argument with a friend about this question :
What is the probability of getting 6 heads in a row from 200 flips ?
I argued it is high probability (significantly bigger than half) while he argued it is low probability.
When i tried to give exact formula i failed so we checked the web were the answer was about 84%, yet he is still not convinced so from this i have two questions:
1) What is the exact formula for $k$ Heads in a row (consecutive) out of $n$ coin flips?
2) (Not a mathematical) How to convince my friend that 6 in a row have high probability ? meaning what is the intuition behind the question ?
probability intuition
edited Dec 29 '18 at 10:58
Did
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
$endgroup$
1
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
3
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
3
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
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@UdayKhanna i watched it, because of that i said it have high probability
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– Ahmad
Dec 29 '18 at 10:13
1
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16
|
show 2 more comments
$begingroup$
A few days ago i had an argument with a friend about this question :
What is the probability of getting 6 heads in a row from 200 flips ?
I argued it is high probability (significantly bigger than half) while he argued it is low probability.
When i tried to give exact formula i failed so we checked the web were the answer was about 84%, yet he is still not convinced so from this i have two questions:
1) What is the exact formula for $k$ Heads in a row (consecutive) out of $n$ coin flips?
2) (Not a mathematical) How to convince my friend that 6 in a row have high probability ? meaning what is the intuition behind the question ?
probability intuition
edited Dec 29 '18 at 10:58
Did
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
$endgroup$
A few days ago i had an argument with a friend about this question :
What is the probability of getting 6 heads in a row from 200 flips ?
I argued it is high probability (significantly bigger than half) while he argued it is low probability.
When i tried to give exact formula i failed so we checked the web were the answer was about 84%, yet he is still not convinced so from this i have two questions:
1) What is the exact formula for $k$ Heads in a row (consecutive) out of $n$ coin flips?
2) (Not a mathematical) How to convince my friend that 6 in a row have high probability ? meaning what is the intuition behind the question ?
probability intuition
probability intuition
edited Dec 29 '18 at 10:58
Did
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
edited Dec 29 '18 at 10:58
Did
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
edited Dec 29 '18 at 10:58
Did
249k23228467
edited Dec 29 '18 at 10:58
Did
249k23228467
edited Dec 29 '18 at 10:58
Did
249k23228467
249k23228467
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
asked Dec 29 '18 at 9:57
AhmadAhmad
2,5771725
2,5771725
1
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
3
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
3
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
$begingroup$
@UdayKhanna i watched it, because of that i said it have high probability
$endgroup$
– Ahmad
Dec 29 '18 at 10:13
1
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16
|
show 2 more comments
1
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
3
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
3
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
$begingroup$
@UdayKhanna i watched it, because of that i said it have high probability
$endgroup$
– Ahmad
Dec 29 '18 at 10:13
1
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16
1
1
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
3
3
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
3
3
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
$begingroup$
@UdayKhanna i watched it, because of that i said it have high probability
$endgroup$
– Ahmad
Dec 29 '18 at 10:13
$begingroup$
@UdayKhanna i watched it, because of that i said it have high probability
$endgroup$
– Ahmad
Dec 29 '18 at 10:13
1
1
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16
|
show 2 more comments
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1
$begingroup$
Maybe a computer simulation would help convince him.
$endgroup$
– littleO
Dec 29 '18 at 10:02
3
$begingroup$
There is a video of numberphile which came out recently which makes the point about how runs of large length seem improbable to people because they don't seem like a "random" enough permutation. Maybe make him watch that and then you could talk about misguided intuitions. numberphile.com/videos/randomness-is-random
$endgroup$
– Uday Khanna
Dec 29 '18 at 10:06
3
$begingroup$
Here's one thought. If you flip a coin 6 times in a row, your chance of "success" (that is, heads on all six tosses) is $p = 1/64$. If you do 33 independent trials of this experiment (for a total of 198 tosses), the probability of failing all 33 trials is $(1 - p)^{33} approx 0.6$. So, $0.4$ is clearly a lower bound on your probability of getting 6 heads in a row at least once when flipping a coin 200 times. It's not a very good lower bound, but it might already be larger than what your friend had in mind.
$endgroup$
– littleO
Dec 29 '18 at 10:11
$begingroup$
@UdayKhanna i watched it, because of that i said it have high probability
$endgroup$
– Ahmad
Dec 29 '18 at 10:13
1
$begingroup$
@littleO great thinking, i do agree that he thinks the chances are almost zero so $0.4$ is great deal, thanks
$endgroup$
– Ahmad
Dec 29 '18 at 10:16