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Biased dice probability question

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.)

probability

share|cite|improve this question

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

New contributor

mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

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.)

probability

share|cite|improve this question

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

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
  

  
  
  
  

add a comment | 

$begingroup$

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.)

probability

share|cite|improve this question

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

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.

$endgroup$

share|cite|improve this question

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

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.

share|cite|improve this question

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

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.

edited 29 mins ago

mathpadawan

2,019422

edited 29 mins ago

mathpadawan

2,019422

asked 33 mins ago

mandymandy

211

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.

asked 33 mins ago

mandymandy

211

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.

share|cite|improve this answer

answered 18 mins ago

peterwhypeterwhy

12.3k21229

$endgroup$

add a comment | 

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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

answered 18 mins ago

peterwhypeterwhy

12.3k21229

$endgroup$

add a comment | 

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

answered 18 mins ago

peterwhypeterwhy

12.3k21229

$endgroup$

add a comment | 

$begingroup$

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

answered 18 mins ago

peterwhypeterwhy

12.3k21229

$endgroup$

share|cite|improve this answer

answered 18 mins ago

peterwhypeterwhy

12.3k21229

answered 18 mins ago

peterwhypeterwhy

12.3k21229

answered 18 mins ago

peterwhypeterwhy

12.3k21229