If I roll a die 25 times what is the probability that the mean outcome is greater than 4?
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You take a fair die to a party and announce that you will roll it 25 times.
You will record each outcome and at the end average the 25 outcomes together to get their arithmetical mean.
You offer a bet: The player puts down 1:if mean exceeds 4, you will give him $21 back, but otherwise, he loses his dollar.Is this a good bet for the player(Using Central limit theorem CTL) ?
I tried to solve check whether it is true:
distributions central-limit-theorem games dice
New contributor
add a comment |
up vote
1
down vote
favorite
You take a fair die to a party and announce that you will roll it 25 times.
You will record each outcome and at the end average the 25 outcomes together to get their arithmetical mean.
You offer a bet: The player puts down 1:if mean exceeds 4, you will give him $21 back, but otherwise, he loses his dollar.Is this a good bet for the player(Using Central limit theorem CTL) ?
I tried to solve check whether it is true:
distributions central-limit-theorem games dice
New contributor
1
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
You take a fair die to a party and announce that you will roll it 25 times.
You will record each outcome and at the end average the 25 outcomes together to get their arithmetical mean.
You offer a bet: The player puts down 1:if mean exceeds 4, you will give him $21 back, but otherwise, he loses his dollar.Is this a good bet for the player(Using Central limit theorem CTL) ?
I tried to solve check whether it is true:
distributions central-limit-theorem games dice
New contributor
You take a fair die to a party and announce that you will roll it 25 times.
You will record each outcome and at the end average the 25 outcomes together to get their arithmetical mean.
You offer a bet: The player puts down 1:if mean exceeds 4, you will give him $21 back, but otherwise, he loses his dollar.Is this a good bet for the player(Using Central limit theorem CTL) ?
I tried to solve check whether it is true:
distributions central-limit-theorem games dice
distributions central-limit-theorem games dice
New contributor
New contributor
edited 13 hours ago
Nick Cox
37.6k478126
37.6k478126
New contributor
asked 20 hours ago
Hassan Ali
91
91
New contributor
New contributor
1
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago
add a comment |
1
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago
1
1
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago
add a comment |
2 Answers
2
active
oldest
votes
up vote
3
down vote
I have some hints for you and I suggest you work from there. I might have made errors as well. If you need more help or find an error, leave a comment.
First we write the probability mass function for a having the result $N$ on a single die:
$$prob(N_i)=1/6,, textwith N=1,2,3,4,5,6$$
This is a Discrete Uniform Distribution. All results are equally likely since the die is fair. Now consider the result of summing the faces of 25 rolls of the die and taking the average. The average $S$ is given by the sum of $N_i$ with $i=1,...25$ over 25 where the $N_i$ are i.i.d. distributed as above.
$$S= fracN_1 + N_2 + ... +N_2525$$
So we ask what is $prob(S>4)$.
For that you first need the probability distribution of $S$. Here is where the hint with the Central Limit Theorem comes in handy. It tells us that we can approximate the density of $prob(S)$ can be approximated as Gaussian with
$$prob(S) sim mathcalN(mu,sigma^2/25)$$
where $mu$ is the expected value of the uniform distribution from above and $sigma^2$ is its variance. You can calculate them using the information on this Wikipedia page.
EDIT
If you do not want to use the Gaussian approximation, have a look at the Bates Distribution as suggested by @wolfies.
Do you think you can take it from here?
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
add a comment |
up vote
3
down vote
You are effectively taking the probability for a single dice roll being larger than 4 (which is indeed $frac13$). But that is a different thing than the mean of several dice rolls being larger than 4 (you can express this also as the sum of $x$ dice rolls being larger than $4x$).
Two dice rolls example
See for instance the possible outcomes of two dice rolls, where only $frac1036 < frac13$ have a mean above 4 (or total above 8)
$$beginarrayc
& 1 & 2 & 3 & 4 & 5 & 6 \
hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \
2 & 3 & 4 & 5 & 6 & 7 & 8 \
3 & 4 & 5 & 6 & 7 & 8 & colorred9 \
4 & 5 & 6 & 7 & 8 & colorred9 & colorred10 \
5 & 6 & 7 & 8 & colorred9 & colorred10 & colorred11\
6 & 7 & 8 & colorred9 & colorred10 & colorred11 & colorred12
endarray$$
More dice rolls
The image below shows how this continues for more dice rolls, by plotting the probability of the sum of dice rolls $X$.
25 dice rolls
Explicit: You can calculate this explicitly by computing a table like above for the two dice rolls, but then for many instead. This site there has already been a question about this (How to easily determine the results distribution for multiple dice?). Such calculation will give you a probability of: $$P(barx>4) = frac18231483546232988166^25 approx 0.0641 > frac121 $$Approximation with normal distribution: In the image above you might note the bell shape curve of the normal distribution. The normal distribution is actually a quite good approximation for the mean of a dice roll (in fact the normal distribution was first described in relation to the approximation of coin flips, where deMoivre used a precursor of the normal distribution to approximated the binomial distribution, and you might see the dice roll as a multivariate generalization of the coin flip).
This is 'what the question wants you to do': Use the normal distribution as an approximation for the mean of dice rolls. (and then use the resulting expression for the normal distribution to compute $P(barx>4)$). The question mentions CLT (the central limit theorem), and when you look that up you may find expressions for "approximating the mean of a sample based on the variance of the distribution" (if one would be pedantic then one could say that this approximation is not exactly the same as the 'central limit theorem', but many people mention/use this term when they employ this type of approximation).
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
I have some hints for you and I suggest you work from there. I might have made errors as well. If you need more help or find an error, leave a comment.
First we write the probability mass function for a having the result $N$ on a single die:
$$prob(N_i)=1/6,, textwith N=1,2,3,4,5,6$$
This is a Discrete Uniform Distribution. All results are equally likely since the die is fair. Now consider the result of summing the faces of 25 rolls of the die and taking the average. The average $S$ is given by the sum of $N_i$ with $i=1,...25$ over 25 where the $N_i$ are i.i.d. distributed as above.
$$S= fracN_1 + N_2 + ... +N_2525$$
So we ask what is $prob(S>4)$.
For that you first need the probability distribution of $S$. Here is where the hint with the Central Limit Theorem comes in handy. It tells us that we can approximate the density of $prob(S)$ can be approximated as Gaussian with
$$prob(S) sim mathcalN(mu,sigma^2/25)$$
where $mu$ is the expected value of the uniform distribution from above and $sigma^2$ is its variance. You can calculate them using the information on this Wikipedia page.
EDIT
If you do not want to use the Gaussian approximation, have a look at the Bates Distribution as suggested by @wolfies.
Do you think you can take it from here?
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
add a comment |
up vote
3
down vote
I have some hints for you and I suggest you work from there. I might have made errors as well. If you need more help or find an error, leave a comment.
First we write the probability mass function for a having the result $N$ on a single die:
$$prob(N_i)=1/6,, textwith N=1,2,3,4,5,6$$
This is a Discrete Uniform Distribution. All results are equally likely since the die is fair. Now consider the result of summing the faces of 25 rolls of the die and taking the average. The average $S$ is given by the sum of $N_i$ with $i=1,...25$ over 25 where the $N_i$ are i.i.d. distributed as above.
$$S= fracN_1 + N_2 + ... +N_2525$$
So we ask what is $prob(S>4)$.
For that you first need the probability distribution of $S$. Here is where the hint with the Central Limit Theorem comes in handy. It tells us that we can approximate the density of $prob(S)$ can be approximated as Gaussian with
$$prob(S) sim mathcalN(mu,sigma^2/25)$$
where $mu$ is the expected value of the uniform distribution from above and $sigma^2$ is its variance. You can calculate them using the information on this Wikipedia page.
EDIT
If you do not want to use the Gaussian approximation, have a look at the Bates Distribution as suggested by @wolfies.
Do you think you can take it from here?
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
add a comment |
up vote
3
down vote
up vote
3
down vote
I have some hints for you and I suggest you work from there. I might have made errors as well. If you need more help or find an error, leave a comment.
First we write the probability mass function for a having the result $N$ on a single die:
$$prob(N_i)=1/6,, textwith N=1,2,3,4,5,6$$
This is a Discrete Uniform Distribution. All results are equally likely since the die is fair. Now consider the result of summing the faces of 25 rolls of the die and taking the average. The average $S$ is given by the sum of $N_i$ with $i=1,...25$ over 25 where the $N_i$ are i.i.d. distributed as above.
$$S= fracN_1 + N_2 + ... +N_2525$$
So we ask what is $prob(S>4)$.
For that you first need the probability distribution of $S$. Here is where the hint with the Central Limit Theorem comes in handy. It tells us that we can approximate the density of $prob(S)$ can be approximated as Gaussian with
$$prob(S) sim mathcalN(mu,sigma^2/25)$$
where $mu$ is the expected value of the uniform distribution from above and $sigma^2$ is its variance. You can calculate them using the information on this Wikipedia page.
EDIT
If you do not want to use the Gaussian approximation, have a look at the Bates Distribution as suggested by @wolfies.
Do you think you can take it from here?
I have some hints for you and I suggest you work from there. I might have made errors as well. If you need more help or find an error, leave a comment.
First we write the probability mass function for a having the result $N$ on a single die:
$$prob(N_i)=1/6,, textwith N=1,2,3,4,5,6$$
This is a Discrete Uniform Distribution. All results are equally likely since the die is fair. Now consider the result of summing the faces of 25 rolls of the die and taking the average. The average $S$ is given by the sum of $N_i$ with $i=1,...25$ over 25 where the $N_i$ are i.i.d. distributed as above.
$$S= fracN_1 + N_2 + ... +N_2525$$
So we ask what is $prob(S>4)$.
For that you first need the probability distribution of $S$. Here is where the hint with the Central Limit Theorem comes in handy. It tells us that we can approximate the density of $prob(S)$ can be approximated as Gaussian with
$$prob(S) sim mathcalN(mu,sigma^2/25)$$
where $mu$ is the expected value of the uniform distribution from above and $sigma^2$ is its variance. You can calculate them using the information on this Wikipedia page.
EDIT
If you do not want to use the Gaussian approximation, have a look at the Bates Distribution as suggested by @wolfies.
Do you think you can take it from here?
edited 18 hours ago
answered 19 hours ago
geo
806
806
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
add a comment |
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
I think we should make him do it without the CLT: the question asks what is the probability -- not what is the approximate probability.
– wolfies
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
@wolfies In that case I suggest Hassan should have a look at the Irwin-Hall distribution. I just assumed the CLT part was part of the assignment.
– geo
19 hours ago
1
1
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
Or the Bates distribution (for the sample mean)
– wolfies
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
@wolfies Thanks, I have edited my answer to include the Bates distribution. I incorrectly suggested the Irwin-Hall.
– geo
18 hours ago
add a comment |
up vote
3
down vote
You are effectively taking the probability for a single dice roll being larger than 4 (which is indeed $frac13$). But that is a different thing than the mean of several dice rolls being larger than 4 (you can express this also as the sum of $x$ dice rolls being larger than $4x$).
Two dice rolls example
See for instance the possible outcomes of two dice rolls, where only $frac1036 < frac13$ have a mean above 4 (or total above 8)
$$beginarrayc
& 1 & 2 & 3 & 4 & 5 & 6 \
hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \
2 & 3 & 4 & 5 & 6 & 7 & 8 \
3 & 4 & 5 & 6 & 7 & 8 & colorred9 \
4 & 5 & 6 & 7 & 8 & colorred9 & colorred10 \
5 & 6 & 7 & 8 & colorred9 & colorred10 & colorred11\
6 & 7 & 8 & colorred9 & colorred10 & colorred11 & colorred12
endarray$$
More dice rolls
The image below shows how this continues for more dice rolls, by plotting the probability of the sum of dice rolls $X$.
25 dice rolls
Explicit: You can calculate this explicitly by computing a table like above for the two dice rolls, but then for many instead. This site there has already been a question about this (How to easily determine the results distribution for multiple dice?). Such calculation will give you a probability of: $$P(barx>4) = frac18231483546232988166^25 approx 0.0641 > frac121 $$Approximation with normal distribution: In the image above you might note the bell shape curve of the normal distribution. The normal distribution is actually a quite good approximation for the mean of a dice roll (in fact the normal distribution was first described in relation to the approximation of coin flips, where deMoivre used a precursor of the normal distribution to approximated the binomial distribution, and you might see the dice roll as a multivariate generalization of the coin flip).
This is 'what the question wants you to do': Use the normal distribution as an approximation for the mean of dice rolls. (and then use the resulting expression for the normal distribution to compute $P(barx>4)$). The question mentions CLT (the central limit theorem), and when you look that up you may find expressions for "approximating the mean of a sample based on the variance of the distribution" (if one would be pedantic then one could say that this approximation is not exactly the same as the 'central limit theorem', but many people mention/use this term when they employ this type of approximation).
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
add a comment |
up vote
3
down vote
You are effectively taking the probability for a single dice roll being larger than 4 (which is indeed $frac13$). But that is a different thing than the mean of several dice rolls being larger than 4 (you can express this also as the sum of $x$ dice rolls being larger than $4x$).
Two dice rolls example
See for instance the possible outcomes of two dice rolls, where only $frac1036 < frac13$ have a mean above 4 (or total above 8)
$$beginarrayc
& 1 & 2 & 3 & 4 & 5 & 6 \
hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \
2 & 3 & 4 & 5 & 6 & 7 & 8 \
3 & 4 & 5 & 6 & 7 & 8 & colorred9 \
4 & 5 & 6 & 7 & 8 & colorred9 & colorred10 \
5 & 6 & 7 & 8 & colorred9 & colorred10 & colorred11\
6 & 7 & 8 & colorred9 & colorred10 & colorred11 & colorred12
endarray$$
More dice rolls
The image below shows how this continues for more dice rolls, by plotting the probability of the sum of dice rolls $X$.
25 dice rolls
Explicit: You can calculate this explicitly by computing a table like above for the two dice rolls, but then for many instead. This site there has already been a question about this (How to easily determine the results distribution for multiple dice?). Such calculation will give you a probability of: $$P(barx>4) = frac18231483546232988166^25 approx 0.0641 > frac121 $$Approximation with normal distribution: In the image above you might note the bell shape curve of the normal distribution. The normal distribution is actually a quite good approximation for the mean of a dice roll (in fact the normal distribution was first described in relation to the approximation of coin flips, where deMoivre used a precursor of the normal distribution to approximated the binomial distribution, and you might see the dice roll as a multivariate generalization of the coin flip).
This is 'what the question wants you to do': Use the normal distribution as an approximation for the mean of dice rolls. (and then use the resulting expression for the normal distribution to compute $P(barx>4)$). The question mentions CLT (the central limit theorem), and when you look that up you may find expressions for "approximating the mean of a sample based on the variance of the distribution" (if one would be pedantic then one could say that this approximation is not exactly the same as the 'central limit theorem', but many people mention/use this term when they employ this type of approximation).
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
add a comment |
up vote
3
down vote
up vote
3
down vote
You are effectively taking the probability for a single dice roll being larger than 4 (which is indeed $frac13$). But that is a different thing than the mean of several dice rolls being larger than 4 (you can express this also as the sum of $x$ dice rolls being larger than $4x$).
Two dice rolls example
See for instance the possible outcomes of two dice rolls, where only $frac1036 < frac13$ have a mean above 4 (or total above 8)
$$beginarrayc
& 1 & 2 & 3 & 4 & 5 & 6 \
hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \
2 & 3 & 4 & 5 & 6 & 7 & 8 \
3 & 4 & 5 & 6 & 7 & 8 & colorred9 \
4 & 5 & 6 & 7 & 8 & colorred9 & colorred10 \
5 & 6 & 7 & 8 & colorred9 & colorred10 & colorred11\
6 & 7 & 8 & colorred9 & colorred10 & colorred11 & colorred12
endarray$$
More dice rolls
The image below shows how this continues for more dice rolls, by plotting the probability of the sum of dice rolls $X$.
25 dice rolls
Explicit: You can calculate this explicitly by computing a table like above for the two dice rolls, but then for many instead. This site there has already been a question about this (How to easily determine the results distribution for multiple dice?). Such calculation will give you a probability of: $$P(barx>4) = frac18231483546232988166^25 approx 0.0641 > frac121 $$Approximation with normal distribution: In the image above you might note the bell shape curve of the normal distribution. The normal distribution is actually a quite good approximation for the mean of a dice roll (in fact the normal distribution was first described in relation to the approximation of coin flips, where deMoivre used a precursor of the normal distribution to approximated the binomial distribution, and you might see the dice roll as a multivariate generalization of the coin flip).
This is 'what the question wants you to do': Use the normal distribution as an approximation for the mean of dice rolls. (and then use the resulting expression for the normal distribution to compute $P(barx>4)$). The question mentions CLT (the central limit theorem), and when you look that up you may find expressions for "approximating the mean of a sample based on the variance of the distribution" (if one would be pedantic then one could say that this approximation is not exactly the same as the 'central limit theorem', but many people mention/use this term when they employ this type of approximation).
You are effectively taking the probability for a single dice roll being larger than 4 (which is indeed $frac13$). But that is a different thing than the mean of several dice rolls being larger than 4 (you can express this also as the sum of $x$ dice rolls being larger than $4x$).
Two dice rolls example
See for instance the possible outcomes of two dice rolls, where only $frac1036 < frac13$ have a mean above 4 (or total above 8)
$$beginarrayc
& 1 & 2 & 3 & 4 & 5 & 6 \
hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \
2 & 3 & 4 & 5 & 6 & 7 & 8 \
3 & 4 & 5 & 6 & 7 & 8 & colorred9 \
4 & 5 & 6 & 7 & 8 & colorred9 & colorred10 \
5 & 6 & 7 & 8 & colorred9 & colorred10 & colorred11\
6 & 7 & 8 & colorred9 & colorred10 & colorred11 & colorred12
endarray$$
More dice rolls
The image below shows how this continues for more dice rolls, by plotting the probability of the sum of dice rolls $X$.
25 dice rolls
Explicit: You can calculate this explicitly by computing a table like above for the two dice rolls, but then for many instead. This site there has already been a question about this (How to easily determine the results distribution for multiple dice?). Such calculation will give you a probability of: $$P(barx>4) = frac18231483546232988166^25 approx 0.0641 > frac121 $$Approximation with normal distribution: In the image above you might note the bell shape curve of the normal distribution. The normal distribution is actually a quite good approximation for the mean of a dice roll (in fact the normal distribution was first described in relation to the approximation of coin flips, where deMoivre used a precursor of the normal distribution to approximated the binomial distribution, and you might see the dice roll as a multivariate generalization of the coin flip).
This is 'what the question wants you to do': Use the normal distribution as an approximation for the mean of dice rolls. (and then use the resulting expression for the normal distribution to compute $P(barx>4)$). The question mentions CLT (the central limit theorem), and when you look that up you may find expressions for "approximating the mean of a sample based on the variance of the distribution" (if one would be pedantic then one could say that this approximation is not exactly the same as the 'central limit theorem', but many people mention/use this term when they employ this type of approximation).
edited 13 hours ago
answered 15 hours ago
Martijn Weterings
11.1k1354
11.1k1354
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
add a comment |
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
I am little confused that you use two dies or two rolls of a single die.In my question we roll single die 25 times
– Hassan Ali
11 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
@HassanAli I use two dies as simplified example for changes that occur when you go from one dice roll to a mean of multiple dice rolls. In this way you can more intuitively get an overview of what happens when you take the mean of 25 dice rolls. The example with the two dice rolls show explicitly how your calculation of $P(barx>4) = 1/3$ is wrong. Note that the answer contains a link to how to easily determine the results distribution for multiple dice (although the aim of your question seems to be that you use the normal distribution)
– Martijn Weterings
10 hours ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
Wetering So can i use similar process to anwer my question because i think the answer can be logical only using the above P(x¯>4)=0.0641 that player shouldn't bet or bet isn't good for player.
– Hassan Ali
1 hour ago
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Hassan Ali is a new contributor. Be nice, and check out our Code of Conduct.
Hassan Ali is a new contributor. Be nice, and check out our Code of Conduct.
Hassan Ali is a new contributor. Be nice, and check out our Code of Conduct.
Hassan Ali is a new contributor. Be nice, and check out our Code of Conduct.
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1
When this is a homework question then you should edit the tags in your question and add 'self-study'.
– Martijn Weterings
13 hours ago
Ok thanks for recomending @Martijn Weterings
– Hassan Ali
11 hours ago