A game of probability
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An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2?
According to my reasoning the answer should be $frac5^46^4$ because we have the options 2,3,4,5 and 6 . Which can be chosen for four trials in $5^4$ ways .
However the correct answer seems to be $frac5^4-4^46^4$ and I canâÂÂt reason it out. Please help me where IâÂÂm missing out
Thank you !
probability
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up vote
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favorite
An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2?
According to my reasoning the answer should be $frac5^46^4$ because we have the options 2,3,4,5 and 6 . Which can be chosen for four trials in $5^4$ ways .
However the correct answer seems to be $frac5^4-4^46^4$ and I canâÂÂt reason it out. Please help me where IâÂÂm missing out
Thank you !
probability
$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2?
According to my reasoning the answer should be $frac5^46^4$ because we have the options 2,3,4,5 and 6 . Which can be chosen for four trials in $5^4$ ways .
However the correct answer seems to be $frac5^4-4^46^4$ and I canâÂÂt reason it out. Please help me where IâÂÂm missing out
Thank you !
probability
An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2?
According to my reasoning the answer should be $frac5^46^4$ because we have the options 2,3,4,5 and 6 . Which can be chosen for four trials in $5^4$ ways .
However the correct answer seems to be $frac5^4-4^46^4$ and I canâÂÂt reason it out. Please help me where IâÂÂm missing out
Thank you !
probability
probability
asked 56 mins ago
Aditi
708314
708314
$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago
add a comment |Â
$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago
$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago
add a comment |Â
3 Answers
3
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oldest
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up vote
3
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accepted
There are $5^4$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $4^4$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $5^4-4^4$ sequences thus have minimum digit exactly 2, as required, and dividing by the $6^4$ rolls in total yields the correct probability.
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
add a comment |Â
up vote
3
down vote
Hint:
Your answer would be good if the question were "What is the probability that the minimum face is not $1$"?
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
add a comment |Â
up vote
0
down vote
You can also calculate the numerator directly: 4*4^3 sequences have a single 2: 4C2*4^2 have two 2's; 4C3*4 sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get 369=5^4-4^4.
New contributor
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
There are $5^4$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $4^4$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $5^4-4^4$ sequences thus have minimum digit exactly 2, as required, and dividing by the $6^4$ rolls in total yields the correct probability.
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
add a comment |Â
up vote
3
down vote
accepted
There are $5^4$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $4^4$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $5^4-4^4$ sequences thus have minimum digit exactly 2, as required, and dividing by the $6^4$ rolls in total yields the correct probability.
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
There are $5^4$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $4^4$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $5^4-4^4$ sequences thus have minimum digit exactly 2, as required, and dividing by the $6^4$ rolls in total yields the correct probability.
There are $5^4$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $4^4$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $5^4-4^4$ sequences thus have minimum digit exactly 2, as required, and dividing by the $6^4$ rolls in total yields the correct probability.
answered 52 mins ago
Parcly Taxel
36.6k137095
36.6k137095
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
add a comment |Â
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
Ohhh I didnâÂÂt know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ?
â Aditi
50 mins ago
1
1
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
@Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum.
â Parcly Taxel
49 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
Ohh now I get it ! Thank you :)
â Aditi
48 mins ago
add a comment |Â
up vote
3
down vote
Hint:
Your answer would be good if the question were "What is the probability that the minimum face is not $1$"?
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
add a comment |Â
up vote
3
down vote
Hint:
Your answer would be good if the question were "What is the probability that the minimum face is not $1$"?
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Hint:
Your answer would be good if the question were "What is the probability that the minimum face is not $1$"?
Hint:
Your answer would be good if the question were "What is the probability that the minimum face is not $1$"?
answered 52 mins ago
ajotatxe
50.7k13185
50.7k13185
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
add a comment |Â
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
Thank you , but how do we deduce that 2 must be one of the outcomes ?
â Aditi
49 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
You can't. Every roll could be fives and sixes (for example). There it is the point.
â ajotatxe
48 mins ago
Thank you for the hint !
â Aditi
45 mins ago
Thank you for the hint !
â Aditi
45 mins ago
add a comment |Â
up vote
0
down vote
You can also calculate the numerator directly: 4*4^3 sequences have a single 2: 4C2*4^2 have two 2's; 4C3*4 sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get 369=5^4-4^4.
New contributor
add a comment |Â
up vote
0
down vote
You can also calculate the numerator directly: 4*4^3 sequences have a single 2: 4C2*4^2 have two 2's; 4C3*4 sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get 369=5^4-4^4.
New contributor
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You can also calculate the numerator directly: 4*4^3 sequences have a single 2: 4C2*4^2 have two 2's; 4C3*4 sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get 369=5^4-4^4.
New contributor
You can also calculate the numerator directly: 4*4^3 sequences have a single 2: 4C2*4^2 have two 2's; 4C3*4 sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get 369=5^4-4^4.
New contributor
New contributor
answered 32 mins ago
G. Thompson
11
11
New contributor
New contributor
add a comment |Â
add a comment |Â
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$(5/6)^4$ is the probability that the minimum face value is $ge 2$.
â ncmathsadist
54 mins ago
@ncmathsadist so do you think that my answer is correct ? Please help
â Aditi
53 mins ago