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.
$begingroup$
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 1 hour ago
mathpadawan
2,019422
asked 1 hour ago
mandymandy
191
New contributor
mandy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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 1 hour ago
mathpadawan
2,019422
asked 1 hour ago
mandymandy
191
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
56 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 1 hour ago
mathpadawan
2,019422
asked 1 hour ago
mandymandy
191
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 1 hour ago
mathpadawan
2,019422
asked 1 hour ago
mandymandy
191
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 1 hour ago
mathpadawan
2,019422
asked 1 hour ago
mandymandy
191
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 1 hour ago
mathpadawan
2,019422
edited 1 hour ago
mathpadawan
2,019422
edited 1 hour ago
mathpadawan
2,019422
2,019422
asked 1 hour ago
mandymandy
191
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 1 hour ago
mandymandy
191
asked 1 hour ago
mandymandy
191
191
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
56 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
56 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
56 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
56 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 55 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
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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 55 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 55 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 55 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 55 mins ago
peterwhypeterwhy
12.3k21229
answered 55 mins ago
peterwhypeterwhy
12.3k21229
answered 55 mins ago
peterwhypeterwhy
12.3k21229
answered 55 mins ago
peterwhypeterwhy
12.3k21229
12.3k21229
add a comment |
add a comment |
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mandy is a new contributor. Be nice, and check out our Code of Conduct.
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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
56 mins ago