Probability Question About Basketball Free Throws and Distributions












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I was trying to solve the following question from the Harvard Stat $110$ course, from the book Introduction to Probability by Blitzstein and Hwang, but could not wrap my head around it. Any and all help is appreciated, especially with part (b):




A certain basketball player practices shooting free throws over and over
again. The shots are independent, with probability p of success.




  • (a) In $n$ shots, what is the expected number of streaks of $7$ consecutive successful shots?
    (Note that, for example, $9$ in a row counts as $3$ streaks.)


  • (b) Now suppose that the player keeps shooting until making $7$ shots in a row for the first time. Let $X$ be the number of shots taken. Show that $E(X) ≤ 7/p^7.$





Hint: Consider the first $7$ trials as a block, then the next $7$ as a block, etc.






probability probability-theory statistics probability-distributions







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    1












    $begingroup$


    I was trying to solve the following question from the Harvard Stat $110$ course, from the book Introduction to Probability by Blitzstein and Hwang, but could not wrap my head around it. Any and all help is appreciated, especially with part (b):




    A certain basketball player practices shooting free throws over and over
    again. The shots are independent, with probability p of success.




    • (a) In $n$ shots, what is the expected number of streaks of $7$ consecutive successful shots?
      (Note that, for example, $9$ in a row counts as $3$ streaks.)


    • (b) Now suppose that the player keeps shooting until making $7$ shots in a row for the first time. Let $X$ be the number of shots taken. Show that $E(X) ≤ 7/p^7.$





    Hint: Consider the first $7$ trials as a block, then the next $7$ as a block, etc.






    probability probability-theory statistics probability-distributions







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      0



      $begingroup$


      I was trying to solve the following question from the Harvard Stat $110$ course, from the book Introduction to Probability by Blitzstein and Hwang, but could not wrap my head around it. Any and all help is appreciated, especially with part (b):




      A certain basketball player practices shooting free throws over and over
      again. The shots are independent, with probability p of success.




      • (a) In $n$ shots, what is the expected number of streaks of $7$ consecutive successful shots?
        (Note that, for example, $9$ in a row counts as $3$ streaks.)


      • (b) Now suppose that the player keeps shooting until making $7$ shots in a row for the first time. Let $X$ be the number of shots taken. Show that $E(X) ≤ 7/p^7.$





      Hint: Consider the first $7$ trials as a block, then the next $7$ as a block, etc.






      probability probability-theory statistics probability-distributions







      share|cite|improve this question











      $endgroup$




      I was trying to solve the following question from the Harvard Stat $110$ course, from the book Introduction to Probability by Blitzstein and Hwang, but could not wrap my head around it. Any and all help is appreciated, especially with part (b):




      A certain basketball player practices shooting free throws over and over
      again. The shots are independent, with probability p of success.




      • (a) In $n$ shots, what is the expected number of streaks of $7$ consecutive successful shots?
        (Note that, for example, $9$ in a row counts as $3$ streaks.)


      • (b) Now suppose that the player keeps shooting until making $7$ shots in a row for the first time. Let $X$ be the number of shots taken. Show that $E(X) ≤ 7/p^7.$





      Hint: Consider the first $7$ trials as a block, then the next $7$ as a block, etc.






      probability probability-theory statistics probability-distributions




      probability probability-theory statistics probability-distributions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Oct 4 '16 at 9:50









      Jimmy R.

      33.1k42257




      33.1k42257










      asked Oct 4 '16 at 5:44









      mathstudentmathstudent

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          2 Answers
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          $begingroup$

          I'm not sure I understand your explanation on (a), but here's my take on (b):



          Let $E(n)$ be the expected number of shots taken until $n$ shots are made in a row. We have the following recursive relationship:
          $$E(n) = E(n-1) + p + (1-p)(1+E(n)).$$
          Simplifying this, we get:
          $$E(n) = frac{1}{p} + frac{E(n-1)}{p}$$
          Knowing $E(0) = 0$, we can prove that $E(7) = frac{1}{p}+frac{1}{p^2}+frac{1}{p^3}+frac{1}{p^4}+frac{1}{p^5}+frac{1}{p^6}+frac{1}{p^7}leq frac{7}{p^7}.$






          share|cite|improve this answer









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            0












            $begingroup$

            (b) has already been answered very well, so here is my take on (a):



            Let us use linearity of expected value.



            Let $I_n$ denote a random variable such that $I_n=1$ if and only if the $n$-th shot was the end of a streak ($I_n=0$ for $n<7$).



            Then the requiered expected value $E(n)=sumlimits_{i=1}^n mathbb{E}[I_n]=sumlimits_{i=1}^n P[I_n=1]=sumlimits_{i=7}^n p^7=(n-6)p^7$






            share|cite|improve this answer









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              2 Answers
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              2 Answers
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              active

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              0












              $begingroup$

              I'm not sure I understand your explanation on (a), but here's my take on (b):



              Let $E(n)$ be the expected number of shots taken until $n$ shots are made in a row. We have the following recursive relationship:
              $$E(n) = E(n-1) + p + (1-p)(1+E(n)).$$
              Simplifying this, we get:
              $$E(n) = frac{1}{p} + frac{E(n-1)}{p}$$
              Knowing $E(0) = 0$, we can prove that $E(7) = frac{1}{p}+frac{1}{p^2}+frac{1}{p^3}+frac{1}{p^4}+frac{1}{p^5}+frac{1}{p^6}+frac{1}{p^7}leq frac{7}{p^7}.$






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                I'm not sure I understand your explanation on (a), but here's my take on (b):



                Let $E(n)$ be the expected number of shots taken until $n$ shots are made in a row. We have the following recursive relationship:
                $$E(n) = E(n-1) + p + (1-p)(1+E(n)).$$
                Simplifying this, we get:
                $$E(n) = frac{1}{p} + frac{E(n-1)}{p}$$
                Knowing $E(0) = 0$, we can prove that $E(7) = frac{1}{p}+frac{1}{p^2}+frac{1}{p^3}+frac{1}{p^4}+frac{1}{p^5}+frac{1}{p^6}+frac{1}{p^7}leq frac{7}{p^7}.$






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  I'm not sure I understand your explanation on (a), but here's my take on (b):



                  Let $E(n)$ be the expected number of shots taken until $n$ shots are made in a row. We have the following recursive relationship:
                  $$E(n) = E(n-1) + p + (1-p)(1+E(n)).$$
                  Simplifying this, we get:
                  $$E(n) = frac{1}{p} + frac{E(n-1)}{p}$$
                  Knowing $E(0) = 0$, we can prove that $E(7) = frac{1}{p}+frac{1}{p^2}+frac{1}{p^3}+frac{1}{p^4}+frac{1}{p^5}+frac{1}{p^6}+frac{1}{p^7}leq frac{7}{p^7}.$






                  share|cite|improve this answer









                  $endgroup$



                  I'm not sure I understand your explanation on (a), but here's my take on (b):



                  Let $E(n)$ be the expected number of shots taken until $n$ shots are made in a row. We have the following recursive relationship:
                  $$E(n) = E(n-1) + p + (1-p)(1+E(n)).$$
                  Simplifying this, we get:
                  $$E(n) = frac{1}{p} + frac{E(n-1)}{p}$$
                  Knowing $E(0) = 0$, we can prove that $E(7) = frac{1}{p}+frac{1}{p^2}+frac{1}{p^3}+frac{1}{p^4}+frac{1}{p^5}+frac{1}{p^6}+frac{1}{p^7}leq frac{7}{p^7}.$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Oct 4 '16 at 22:00









                  trang1618trang1618

                  1,435214




                  1,435214























                      0












                      $begingroup$

                      (b) has already been answered very well, so here is my take on (a):



                      Let us use linearity of expected value.



                      Let $I_n$ denote a random variable such that $I_n=1$ if and only if the $n$-th shot was the end of a streak ($I_n=0$ for $n<7$).



                      Then the requiered expected value $E(n)=sumlimits_{i=1}^n mathbb{E}[I_n]=sumlimits_{i=1}^n P[I_n=1]=sumlimits_{i=7}^n p^7=(n-6)p^7$






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        (b) has already been answered very well, so here is my take on (a):



                        Let us use linearity of expected value.



                        Let $I_n$ denote a random variable such that $I_n=1$ if and only if the $n$-th shot was the end of a streak ($I_n=0$ for $n<7$).



                        Then the requiered expected value $E(n)=sumlimits_{i=1}^n mathbb{E}[I_n]=sumlimits_{i=1}^n P[I_n=1]=sumlimits_{i=7}^n p^7=(n-6)p^7$






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          (b) has already been answered very well, so here is my take on (a):



                          Let us use linearity of expected value.



                          Let $I_n$ denote a random variable such that $I_n=1$ if and only if the $n$-th shot was the end of a streak ($I_n=0$ for $n<7$).



                          Then the requiered expected value $E(n)=sumlimits_{i=1}^n mathbb{E}[I_n]=sumlimits_{i=1}^n P[I_n=1]=sumlimits_{i=7}^n p^7=(n-6)p^7$






                          share|cite|improve this answer









                          $endgroup$



                          (b) has already been answered very well, so here is my take on (a):



                          Let us use linearity of expected value.



                          Let $I_n$ denote a random variable such that $I_n=1$ if and only if the $n$-th shot was the end of a streak ($I_n=0$ for $n<7$).



                          Then the requiered expected value $E(n)=sumlimits_{i=1}^n mathbb{E}[I_n]=sumlimits_{i=1}^n P[I_n=1]=sumlimits_{i=7}^n p^7=(n-6)p^7$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Sep 6 '17 at 18:21









                          SergSerg

                          574315




                          574315






























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