Biased dice probability question
$begingroup$
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
edited 41 mins ago
mathpadawan
2,019422
asked 44 mins ago
mandymandy
211
New contributor
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$endgroup$
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
edited 41 mins ago
mathpadawan
2,019422
asked 44 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
31 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
edited 41 mins ago
mathpadawan
2,019422
asked 44 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.
$endgroup$
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
probability
edited 41 mins ago
mathpadawan
2,019422
asked 44 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 41 mins ago
mathpadawan
2,019422
asked 44 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 41 mins ago
mathpadawan
2,019422
edited 41 mins ago
mathpadawan
2,019422
edited 41 mins ago
mathpadawan
2,019422
2,019422
asked 44 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 44 mins ago
mandymandy
211
asked 44 mins ago
mandymandy
211
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.
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.
mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
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
31 mins ago
add a comment |
$begingroup$
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
31 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
31 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
31 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 30 mins ago
peterwhypeterwhy
12.3k21229
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
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.
answered 30 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.
answered 30 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.
answered 30 mins ago
peterwhypeterwhy
12.3k21229
$endgroup$
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.
answered 30 mins ago
peterwhypeterwhy
12.3k21229
answered 30 mins ago
peterwhypeterwhy
12.3k21229
answered 30 mins ago
peterwhypeterwhy
12.3k21229
answered 30 mins ago
peterwhypeterwhy
12.3k21229
12.3k21229
add a comment |
add a comment |
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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
31 mins ago