Rolling a fair 6 sided die k times











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The question:



Suppose we roll a fair 6 sided die with the number [1,6] written on them. After the first die roll we roll the die $k$ times where $k$ is the number on the first die roll. The number of points you score is the sum of the face-values on all die rolls (including the first). What is the expected number of points you will score?



Attempt:



There are six different cases for the first roll





  • Case 1: Roll 1





    • $P(D_1=1) =frac{1}{6}$

    • The expected number on the next roll is $E(D_2)=frac{1}{6}times(1+2+3+4+5+6)=3.5$


    • $E(S_1) = 3.5+1=4.5$ points





  • Case 2: Roll 2




    • We know what the expected value of one roll is, and since rolling the die is independent we can use the previous expected value for the next roll.


    • $E(S_2) = (3.5times 2)+2=9$ points





  • Case 3: Roll 3





    • $E(S_3) = (3.5times 3)+3=13.5$ points




  • Case 4: Roll 4





    • $E(S_4) = (3.5times 4)+4=18$ points




  • Case 5: Roll 5





    • $E(S_5) = (3.5times 5)+5=22.5$ points




  • Case 6: Roll 6





    • $E(S_6) = (3.5times 6)+6=27$ points




Therefore, the expected number of points scored is



$=frac{1}{6}times(4.5+9+13.5+18+22.5+27)=15.75$ points










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  • 1




    I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
    – gd1035
    Nov 16 at 15:49






  • 1




    I think your attempt is correct. That is how I would have tried it anyways.
    – Sauhard Sharma
    Nov 16 at 16:22















up vote
1
down vote

favorite












The question:



Suppose we roll a fair 6 sided die with the number [1,6] written on them. After the first die roll we roll the die $k$ times where $k$ is the number on the first die roll. The number of points you score is the sum of the face-values on all die rolls (including the first). What is the expected number of points you will score?



Attempt:



There are six different cases for the first roll





  • Case 1: Roll 1





    • $P(D_1=1) =frac{1}{6}$

    • The expected number on the next roll is $E(D_2)=frac{1}{6}times(1+2+3+4+5+6)=3.5$


    • $E(S_1) = 3.5+1=4.5$ points





  • Case 2: Roll 2




    • We know what the expected value of one roll is, and since rolling the die is independent we can use the previous expected value for the next roll.


    • $E(S_2) = (3.5times 2)+2=9$ points





  • Case 3: Roll 3





    • $E(S_3) = (3.5times 3)+3=13.5$ points




  • Case 4: Roll 4





    • $E(S_4) = (3.5times 4)+4=18$ points




  • Case 5: Roll 5





    • $E(S_5) = (3.5times 5)+5=22.5$ points




  • Case 6: Roll 6





    • $E(S_6) = (3.5times 6)+6=27$ points




Therefore, the expected number of points scored is



$=frac{1}{6}times(4.5+9+13.5+18+22.5+27)=15.75$ points










share|cite|improve this question


















  • 1




    I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
    – gd1035
    Nov 16 at 15:49






  • 1




    I think your attempt is correct. That is how I would have tried it anyways.
    – Sauhard Sharma
    Nov 16 at 16:22













up vote
1
down vote

favorite









up vote
1
down vote

favorite











The question:



Suppose we roll a fair 6 sided die with the number [1,6] written on them. After the first die roll we roll the die $k$ times where $k$ is the number on the first die roll. The number of points you score is the sum of the face-values on all die rolls (including the first). What is the expected number of points you will score?



Attempt:



There are six different cases for the first roll





  • Case 1: Roll 1





    • $P(D_1=1) =frac{1}{6}$

    • The expected number on the next roll is $E(D_2)=frac{1}{6}times(1+2+3+4+5+6)=3.5$


    • $E(S_1) = 3.5+1=4.5$ points





  • Case 2: Roll 2




    • We know what the expected value of one roll is, and since rolling the die is independent we can use the previous expected value for the next roll.


    • $E(S_2) = (3.5times 2)+2=9$ points





  • Case 3: Roll 3





    • $E(S_3) = (3.5times 3)+3=13.5$ points




  • Case 4: Roll 4





    • $E(S_4) = (3.5times 4)+4=18$ points




  • Case 5: Roll 5





    • $E(S_5) = (3.5times 5)+5=22.5$ points




  • Case 6: Roll 6





    • $E(S_6) = (3.5times 6)+6=27$ points




Therefore, the expected number of points scored is



$=frac{1}{6}times(4.5+9+13.5+18+22.5+27)=15.75$ points










share|cite|improve this question













The question:



Suppose we roll a fair 6 sided die with the number [1,6] written on them. After the first die roll we roll the die $k$ times where $k$ is the number on the first die roll. The number of points you score is the sum of the face-values on all die rolls (including the first). What is the expected number of points you will score?



Attempt:



There are six different cases for the first roll





  • Case 1: Roll 1





    • $P(D_1=1) =frac{1}{6}$

    • The expected number on the next roll is $E(D_2)=frac{1}{6}times(1+2+3+4+5+6)=3.5$


    • $E(S_1) = 3.5+1=4.5$ points





  • Case 2: Roll 2




    • We know what the expected value of one roll is, and since rolling the die is independent we can use the previous expected value for the next roll.


    • $E(S_2) = (3.5times 2)+2=9$ points





  • Case 3: Roll 3





    • $E(S_3) = (3.5times 3)+3=13.5$ points




  • Case 4: Roll 4





    • $E(S_4) = (3.5times 4)+4=18$ points




  • Case 5: Roll 5





    • $E(S_5) = (3.5times 5)+5=22.5$ points




  • Case 6: Roll 6





    • $E(S_6) = (3.5times 6)+6=27$ points




Therefore, the expected number of points scored is



$=frac{1}{6}times(4.5+9+13.5+18+22.5+27)=15.75$ points







probability combinatorics






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asked Nov 16 at 15:41









Arthur Green

656




656








  • 1




    I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
    – gd1035
    Nov 16 at 15:49






  • 1




    I think your attempt is correct. That is how I would have tried it anyways.
    – Sauhard Sharma
    Nov 16 at 16:22














  • 1




    I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
    – gd1035
    Nov 16 at 15:49






  • 1




    I think your attempt is correct. That is how I would have tried it anyways.
    – Sauhard Sharma
    Nov 16 at 16:22








1




1




I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
– gd1035
Nov 16 at 15:49




I think this is a law of total expectation problem, $E[X]=E[E(X|K)]$. Where $K$ is the face of the first roll, and $X$ is the sum of the $K$ rolls
– gd1035
Nov 16 at 15:49




1




1




I think your attempt is correct. That is how I would have tried it anyways.
– Sauhard Sharma
Nov 16 at 16:22




I think your attempt is correct. That is how I would have tried it anyways.
– Sauhard Sharma
Nov 16 at 16:22










1 Answer
1






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1
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accepted










The expected value of a sum of random number $N$ of iid random variables $X_i$ is
$$Eleft[sum_{i=1}^N X_iright]=E[N]E[X_i]$$
In your case you add $E[N]$, so the answer is
$$E[N]E[X_i]+E[N]=3.5cdot 3.5+3.5 =4.5cdot 3.5 = 15.75$$






share|cite|improve this answer





















  • Why do we add $E[N]$ to the summation?
    – Arthur Green
    Nov 16 at 19:13








  • 1




    Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
    – kludg
    Nov 16 at 20:05











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

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active

oldest

votes








up vote
1
down vote



accepted










The expected value of a sum of random number $N$ of iid random variables $X_i$ is
$$Eleft[sum_{i=1}^N X_iright]=E[N]E[X_i]$$
In your case you add $E[N]$, so the answer is
$$E[N]E[X_i]+E[N]=3.5cdot 3.5+3.5 =4.5cdot 3.5 = 15.75$$






share|cite|improve this answer





















  • Why do we add $E[N]$ to the summation?
    – Arthur Green
    Nov 16 at 19:13








  • 1




    Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
    – kludg
    Nov 16 at 20:05















up vote
1
down vote



accepted










The expected value of a sum of random number $N$ of iid random variables $X_i$ is
$$Eleft[sum_{i=1}^N X_iright]=E[N]E[X_i]$$
In your case you add $E[N]$, so the answer is
$$E[N]E[X_i]+E[N]=3.5cdot 3.5+3.5 =4.5cdot 3.5 = 15.75$$






share|cite|improve this answer





















  • Why do we add $E[N]$ to the summation?
    – Arthur Green
    Nov 16 at 19:13








  • 1




    Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
    – kludg
    Nov 16 at 20:05













up vote
1
down vote



accepted







up vote
1
down vote



accepted






The expected value of a sum of random number $N$ of iid random variables $X_i$ is
$$Eleft[sum_{i=1}^N X_iright]=E[N]E[X_i]$$
In your case you add $E[N]$, so the answer is
$$E[N]E[X_i]+E[N]=3.5cdot 3.5+3.5 =4.5cdot 3.5 = 15.75$$






share|cite|improve this answer












The expected value of a sum of random number $N$ of iid random variables $X_i$ is
$$Eleft[sum_{i=1}^N X_iright]=E[N]E[X_i]$$
In your case you add $E[N]$, so the answer is
$$E[N]E[X_i]+E[N]=3.5cdot 3.5+3.5 =4.5cdot 3.5 = 15.75$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 16 at 17:20









kludg

1,159611




1,159611












  • Why do we add $E[N]$ to the summation?
    – Arthur Green
    Nov 16 at 19:13








  • 1




    Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
    – kludg
    Nov 16 at 20:05


















  • Why do we add $E[N]$ to the summation?
    – Arthur Green
    Nov 16 at 19:13








  • 1




    Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
    – kludg
    Nov 16 at 20:05
















Why do we add $E[N]$ to the summation?
– Arthur Green
Nov 16 at 19:13






Why do we add $E[N]$ to the summation?
– Arthur Green
Nov 16 at 19:13






1




1




Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
– kludg
Nov 16 at 20:05




Because you include the 1st (actually zeroth) roll which determines $N$ into the sum, and because of linearity of expectations it adds $E[N]$.
– kludg
Nov 16 at 20:05


















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