Biased dice probability question Announcing the arrival of Valued Associate #679: Cesar...

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Biased dice probability question



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Probability of dice thrownDice and probabilityDetermine whether the dice is biased based on 10 rollsProbability of events with biased diceProbability of biased diceProbability on biased diceProbability of rolling 2 and 3 numbers in a sequence when rolling 3, 6 sided diceDice probability helpProbability of an “at least” QuestionProbability of biased die.












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$begingroup$


A biased six sided dice is rolled twice. Show that the probability that the two results are the same is at least $frac{1}{6}$.
(Hint: $(p_1 − a)^2 + . . . + (p_6 − a)^2 ≥ 0$ and choose suitable
$p_1, . . . , p_6$, a.)










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  • $begingroup$
    Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
    $endgroup$
    – Lorenzo
    20 mins ago
















4












$begingroup$


A biased six sided dice is rolled twice. Show that the probability that the two results are the same is at least $frac{1}{6}$.
(Hint: $(p_1 − a)^2 + . . . + (p_6 − a)^2 ≥ 0$ and choose suitable
$p_1, . . . , p_6$, a.)










share|cite|improve this question









New contributor




mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
    $endgroup$
    – Lorenzo
    20 mins ago














4












4








4


2



$begingroup$


A biased six sided dice is rolled twice. Show that the probability that the two results are the same is at least $frac{1}{6}$.
(Hint: $(p_1 − a)^2 + . . . + (p_6 − a)^2 ≥ 0$ and choose suitable
$p_1, . . . , p_6$, a.)










share|cite|improve this question









New contributor




mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




A biased six sided dice is rolled twice. Show that the probability that the two results are the same is at least $frac{1}{6}$.
(Hint: $(p_1 − a)^2 + . . . + (p_6 − a)^2 ≥ 0$ and choose suitable
$p_1, . . . , p_6$, a.)







probability






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mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 29 mins ago









mathpadawan

2,019422




2,019422






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asked 33 mins ago









mandymandy

211




211




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mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • $begingroup$
    Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
    $endgroup$
    – Lorenzo
    20 mins ago


















  • $begingroup$
    Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
    $endgroup$
    – Lorenzo
    20 mins ago
















$begingroup$
Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
$endgroup$
– Lorenzo
20 mins ago




$begingroup$
Hint: First try to show that if a coin is flipped twice, the probability that the two results are the same is at least $1/2$. This will help you figure out what to choose as $a$.
$endgroup$
– Lorenzo
20 mins ago










1 Answer
1






active

oldest

votes


















4












$begingroup$

Let $p_i$ be the probability of rolling $i$. Then $sum_{i=1}^6 p_i = 1$.



By Cauchy-Schwarz inequality,



$$begin{align*}
left(sum_{i=1}^6 1^2right) left(sum_{i=1}^6 p_i^2right) &ge
left(sum_{i=1}^6 1p_iright)^2\
6left(sum_{i=1}^6 p_i^2right) &ge 1\
sum_{i=1}^6 p_i^2 &ge frac16end{align*}$$



Equality holds when all the $p_i$ are the same, i.e. when the die is unbiased.






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    1 Answer
    1






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    active

    oldest

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    active

    oldest

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    4












    $begingroup$

    Let $p_i$ be the probability of rolling $i$. Then $sum_{i=1}^6 p_i = 1$.



    By Cauchy-Schwarz inequality,



    $$begin{align*}
    left(sum_{i=1}^6 1^2right) left(sum_{i=1}^6 p_i^2right) &ge
    left(sum_{i=1}^6 1p_iright)^2\
    6left(sum_{i=1}^6 p_i^2right) &ge 1\
    sum_{i=1}^6 p_i^2 &ge frac16end{align*}$$



    Equality holds when all the $p_i$ are the same, i.e. when the die is unbiased.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      Let $p_i$ be the probability of rolling $i$. Then $sum_{i=1}^6 p_i = 1$.



      By Cauchy-Schwarz inequality,



      $$begin{align*}
      left(sum_{i=1}^6 1^2right) left(sum_{i=1}^6 p_i^2right) &ge
      left(sum_{i=1}^6 1p_iright)^2\
      6left(sum_{i=1}^6 p_i^2right) &ge 1\
      sum_{i=1}^6 p_i^2 &ge frac16end{align*}$$



      Equality holds when all the $p_i$ are the same, i.e. when the die is unbiased.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        Let $p_i$ be the probability of rolling $i$. Then $sum_{i=1}^6 p_i = 1$.



        By Cauchy-Schwarz inequality,



        $$begin{align*}
        left(sum_{i=1}^6 1^2right) left(sum_{i=1}^6 p_i^2right) &ge
        left(sum_{i=1}^6 1p_iright)^2\
        6left(sum_{i=1}^6 p_i^2right) &ge 1\
        sum_{i=1}^6 p_i^2 &ge frac16end{align*}$$



        Equality holds when all the $p_i$ are the same, i.e. when the die is unbiased.






        share|cite|improve this answer









        $endgroup$



        Let $p_i$ be the probability of rolling $i$. Then $sum_{i=1}^6 p_i = 1$.



        By Cauchy-Schwarz inequality,



        $$begin{align*}
        left(sum_{i=1}^6 1^2right) left(sum_{i=1}^6 p_i^2right) &ge
        left(sum_{i=1}^6 1p_iright)^2\
        6left(sum_{i=1}^6 p_i^2right) &ge 1\
        sum_{i=1}^6 p_i^2 &ge frac16end{align*}$$



        Equality holds when all the $p_i$ are the same, i.e. when the die is unbiased.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 18 mins ago









        peterwhypeterwhy

        12.3k21229




        12.3k21229






















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