Probability of a single trial within binomial experiment vs. stand-alone bernoulli experiment












2














When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. Now, suppose I flipped a fair coin three times and got each time a head (head-head-head). Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail. But each flip is an independent event - the coin does not know what came out last time.



How this paradox is resolved?



Many thanks!






probability binomial-distribution paradoxes







share|cite|improve this question




















  • 3




    I see no paradox, given trials are independent.
    – coffeemath
    2 hours ago
















2














When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. Now, suppose I flipped a fair coin three times and got each time a head (head-head-head). Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail. But each flip is an independent event - the coin does not know what came out last time.



How this paradox is resolved?



Many thanks!






probability binomial-distribution paradoxes







share|cite|improve this question




















  • 3




    I see no paradox, given trials are independent.
    – coffeemath
    2 hours ago














2












2








2


1





When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. Now, suppose I flipped a fair coin three times and got each time a head (head-head-head). Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail. But each flip is an independent event - the coin does not know what came out last time.



How this paradox is resolved?



Many thanks!






probability binomial-distribution paradoxes







share|cite|improve this question















When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. Now, suppose I flipped a fair coin three times and got each time a head (head-head-head). Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail. But each flip is an independent event - the coin does not know what came out last time.



How this paradox is resolved?



Many thanks!






probability binomial-distribution paradoxes




probability binomial-distribution paradoxes






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago









Eevee Trainer

3,708326




3,708326










asked 2 hours ago









John

856




856








  • 3




    I see no paradox, given trials are independent.
    – coffeemath
    2 hours ago














  • 3




    I see no paradox, given trials are independent.
    – coffeemath
    2 hours ago








3




3




I see no paradox, given trials are independent.
– coffeemath
2 hours ago




I see no paradox, given trials are independent.
– coffeemath
2 hours ago










3 Answers
3






active

oldest

votes


















2














If you flipped a fair coin three times and got heads each time, then on the next toss your probability of getting heads is $1/2$. The fact that you just flipped three heads in a row is irrelevant. There is no paradox.






share|cite|improve this answer





















  • But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
    – John
    1 hour ago










  • @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
    – drhab
    1 hour ago



















2















Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail.




This marks the error.



Granted, there is no paradox - your error is known as the gambler's fallacy (or Monte Carlo fallacy). By assuming that tails is more likely, you assume that the trials are not independent even though they are. The fallacy is then that you feel the events are dependent, when they're not.



This was actually touched on in a recent Numberphile video somewhat: we think that streaks or heads of tails is nonrandom, i.e. would be a hint or sign of dependence or bias, when really they're not.



So there is no paradox, just bad intuition because of how human intuition tends to work.






share|cite|improve this answer































    1















    "...so I can expect the on fourth throw to have higher chances to get finally a tail"




    By independence there is no reason at all for expecting that, so there is no paradox.






    share|cite|improve this answer





















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2














      If you flipped a fair coin three times and got heads each time, then on the next toss your probability of getting heads is $1/2$. The fact that you just flipped three heads in a row is irrelevant. There is no paradox.






      share|cite|improve this answer





















      • But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
        – John
        1 hour ago










      • @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
        – drhab
        1 hour ago
















      2














      If you flipped a fair coin three times and got heads each time, then on the next toss your probability of getting heads is $1/2$. The fact that you just flipped three heads in a row is irrelevant. There is no paradox.






      share|cite|improve this answer





















      • But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
        – John
        1 hour ago










      • @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
        – drhab
        1 hour ago














      2












      2








      2






      If you flipped a fair coin three times and got heads each time, then on the next toss your probability of getting heads is $1/2$. The fact that you just flipped three heads in a row is irrelevant. There is no paradox.






      share|cite|improve this answer












      If you flipped a fair coin three times and got heads each time, then on the next toss your probability of getting heads is $1/2$. The fact that you just flipped three heads in a row is irrelevant. There is no paradox.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered 2 hours ago









      littleO

      28.9k644106




      28.9k644106












      • But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
        – John
        1 hour ago










      • @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
        – drhab
        1 hour ago


















      • But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
        – John
        1 hour ago










      • @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
        – drhab
        1 hour ago
















      But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
      – John
      1 hour ago




      But what if I look retrospectively on the whole sequence? If I did get 4 gets, I would said that how lucky I was to get 4 out of 4 heads. But if I received fourth tail, I would say that 3 out of 4 heads is not so low chances.
      – John
      1 hour ago












      @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
      – drhab
      1 hour ago




      @John The probability on getting $4$ heads by $4$ throws is exactly the same as getting at first $3$ heads and then as last a tail. Also in that case you can say: "how lucky I was to get $3$ heads at first and then a tail as last."
      – drhab
      1 hour ago











      2















      Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail.




      This marks the error.



      Granted, there is no paradox - your error is known as the gambler's fallacy (or Monte Carlo fallacy). By assuming that tails is more likely, you assume that the trials are not independent even though they are. The fallacy is then that you feel the events are dependent, when they're not.



      This was actually touched on in a recent Numberphile video somewhat: we think that streaks or heads of tails is nonrandom, i.e. would be a hint or sign of dependence or bias, when really they're not.



      So there is no paradox, just bad intuition because of how human intuition tends to work.






      share|cite|improve this answer




























        2















        Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail.




        This marks the error.



        Granted, there is no paradox - your error is known as the gambler's fallacy (or Monte Carlo fallacy). By assuming that tails is more likely, you assume that the trials are not independent even though they are. The fallacy is then that you feel the events are dependent, when they're not.



        This was actually touched on in a recent Numberphile video somewhat: we think that streaks or heads of tails is nonrandom, i.e. would be a hint or sign of dependence or bias, when really they're not.



        So there is no paradox, just bad intuition because of how human intuition tends to work.






        share|cite|improve this answer


























          2












          2








          2







          Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail.




          This marks the error.



          Granted, there is no paradox - your error is known as the gambler's fallacy (or Monte Carlo fallacy). By assuming that tails is more likely, you assume that the trials are not independent even though they are. The fallacy is then that you feel the events are dependent, when they're not.



          This was actually touched on in a recent Numberphile video somewhat: we think that streaks or heads of tails is nonrandom, i.e. would be a hint or sign of dependence or bias, when really they're not.



          So there is no paradox, just bad intuition because of how human intuition tends to work.






          share|cite|improve this answer















          Intuitively, the head cannot come out “head” all the time, so I can expect the on fourth throw to have higher chances to get finally a tail.




          This marks the error.



          Granted, there is no paradox - your error is known as the gambler's fallacy (or Monte Carlo fallacy). By assuming that tails is more likely, you assume that the trials are not independent even though they are. The fallacy is then that you feel the events are dependent, when they're not.



          This was actually touched on in a recent Numberphile video somewhat: we think that streaks or heads of tails is nonrandom, i.e. would be a hint or sign of dependence or bias, when really they're not.



          So there is no paradox, just bad intuition because of how human intuition tends to work.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 1 hour ago

























          answered 2 hours ago









          Eevee Trainer

          3,708326




          3,708326























              1















              "...so I can expect the on fourth throw to have higher chances to get finally a tail"




              By independence there is no reason at all for expecting that, so there is no paradox.






              share|cite|improve this answer


























                1















                "...so I can expect the on fourth throw to have higher chances to get finally a tail"




                By independence there is no reason at all for expecting that, so there is no paradox.






                share|cite|improve this answer
























                  1












                  1








                  1







                  "...so I can expect the on fourth throw to have higher chances to get finally a tail"




                  By independence there is no reason at all for expecting that, so there is no paradox.






                  share|cite|improve this answer













                  "...so I can expect the on fourth throw to have higher chances to get finally a tail"




                  By independence there is no reason at all for expecting that, so there is no paradox.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  drhab

                  97.4k544128




                  97.4k544128






























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