Math: discrete probability distribution

[masterchiefo]

Homework Statement

We ask a person to taste 18 biscuits , 8 made to butter ( the other 10 are made to margarine ) , and to identify 8 butter cookies . He does not know the exact number of butter cookies . As he sees no difference , he randomly selects those he claims to be butter . Y = the number of butter cookies correctly identified.

Indicate , the variance of the variable Y.

Homework Equations

The Attempt at a Solution

My work: First thing I need to make sure, do I use hypergeometric formulas or binomial or Poisson? My guess is Hypergeometric. Is that right ?

N = 18
n=8
p= 8/18
q= 1-(8/18)

I then entered those numbers into the variance formula but I do not get the correct answer. what is wrong?
which is variance= npq*(1-n/N)

thank you

 

[andrewkirk]

variance= npq*(1-n/N)

That is not the variance of a hypergeometric distribution. Other than the last term it looks like the variance of a binomial.
Get the correct formula for variance of a hypergeometric and you'll get the correct answer. If you don't have your text handy, wikipedia has a formula.
By the way, how did you eliminate Poisson and binomial as possibilities? It can be done in an objective fashion - guesses should not be necessary?

 

[masterchiefo]

That is not the variance of a hypergeometric distribution. Other than the last term it looks like the variance of a binomial.
Get the correct formula for variance of a hypergeometric and you'll get the correct answer. If you don't have your text handy, wikipedia has a formula.
By the way, how did you eliminate Poisson and binomial as possibilities? It can be done in an objective fashion - guesses should not be necessary?

This is the correct variance formula. I just simplified the last part.

You can also write it as: variance= npq*((N-n)/N-1)
Binomial is only variance=npq

It still does not work. Something is wrong.
This part right here makes it confusing. Because if he knew the exact number of butter cookies then what I did is good. But since he does not know the exact number there is something I have to change and I don't know what it is.
(He does not know the exact number of butter cookies .)

 

[andrewkirk]

You can also write it as: variance= npq*((N-n)/N-1)

That is different from what is written in the OP. But neither is quite correct. The denominator in the last term needs to be (N-1).
Inserting two extra brackets into what you wrote in post 3 will make it correct.

 

[masterchiefo]

That is different from what is written in the OP. But neither is quite correct. The denominator in the last term needs to be (N-1).
Inserting two extra brackets into what you wrote in post 3 will make it correct.

variance= npq*((N-n)/(N-1))

I have used the variance formula over hundred times, I use it correctly in my calculator,

I am more worried about those info:
N = 18
n=8
p= 8/18
q= 1-(8/18)

Because something is wrong with that.============================================================

Look this is the example I did before.

We ask a person to taste 18 biscuits , 8 made to butter ( the other 10 are made to margarine ) , and to identify 8 butter cookies . He does know the exact number of butter cookies . As he sees no difference , he randomly selects those he claims to be butter . Y = the number of butter cookies correctly identified.

In this particular problem, he DOES know the exact number of butter cookies, so I can use those info:

N = 18
n=8
p= 8/18
q= 1-(8/18)
and for this problem the variance is correct when I calculate it.But my problem is that he DOES not know the exact number, so I need to change something but I do not know what.

 

[andrewkirk]

variance= npq*((N-n)/(N-1))

That's correct, and your values of n, p, q, N look correct too.
What number did you get for the variance? I get 1.162. What answer does your textbook suggest?

But my problem is that he DOES not know the exact number, so I need to change something but I do not know what.

Whether he knows the number of butter cookies doesn't affect the calc. All that matters is how many cookies he claims are butter. It seemed implicit in the OP that that number was 8. If it is not 8 then the problem cannot be done without more information being supplied about how he chooses how many cookies to label as butter. A distribution of the number of cookies he labels is needed. Different distributions will give different answers.

 

[masterchiefo]

That's correct, and your values of n, p, q, N look correct too.
What number did you get for the variance? I get 1.162. What answer does your textbook suggest?

Whether he knows the number of butter cookies doesn't affect the calc. All that matters is how many cookies he claims are butter. It seemed implicit in the OP that that number was 8. If it is not 8 then the problem cannot be done without more information being supplied about how he chooses how many cookies to label as butter. A distribution of the number of cookies he labels is needed. Different distributions will give different answers.

It can't be correct because this is 2 different Problem with different answer.

#1 Problem is when he does not know the exact amount.
I don't have the correct answer for this one, but it is not 1.1612, teacher said in class that both problem will have a different answer.

#2 Problem is when he does know the exact amount.
Correct answer for this one = 1.1612

 

[masterchiefo]

That's correct, and your values of n, p, q, N look correct too.
What number did you get for the variance? I get 1.162. What answer does your textbook suggest?

Whether he knows the number of butter cookies doesn't affect the calc. All that matters is how many cookies he claims are butter. It seemed implicit in the OP that that number was 8. If it is not 8 then the problem cannot be done without more information being supplied about how he chooses how many cookies to label as butter. A distribution of the number of cookies he labels is needed. Different distributions will give different answers.

n cannot be 8 because the person does not know how many butter there is and it does not say how many he will pick. It says (he randomly selects those he claims to be butter .) it could be 1 to 18...it doesn't say how many he selects.

So how many will he select? how many of those are butter?

 

[Ray Vickson]

Homework Statement

We ask a person to taste 18 biscuits , 8 made to butter ( the other 10 are made to margarine ) , and to identify 8 butter cookies . He does not know the exact number of butter cookies . As he sees no difference , he randomly selects those he claims to be butter . Y = the number of butter cookies correctly identified.

Indicate , the variance of the variable Y.

Homework Equations

The Attempt at a Solution

My work: First thing I need to make sure, do I use hypergeometric formulas or binomial or Poisson? My guess is Hypergeometric. Is that right ?

N = 18
n=8
p= 8/18
q= 1-(8/18)

I then entered those numbers into the variance formula but I do not get the correct answer. what is wrong?
which is variance= npq*(1-n/N)

thank you

If I understand your problem, you have a group of 18 cookies, 8 made with butter (not TO butter!) and 10 made with margarine. A man tests all 18, but simply guesses which are the butter cookies. In other words, his probability of saying "this is a butter cookie" is 1/2 for each of the 18 cookies tasted with all guesses being independent. Is that what you mean? If so, the number of "butter" guesses is binomial with parameters n = 18 and p = 0.5. If X of his guesses are "butter", some of these may be truly butter cookies and some not; the number of "butter" guesses that are truly "butter" is Y.

Is that a correct good description of your problem? If so, it is not a simple hypergeometric or a simple binomial problem. Can you see why? Can you see how conditional probabilities come into play?

 

[masterchiefo]

n cannot be 8 because the person does not know how many butter there is and it does not say how many he will pick. It says (he randomly selects those he claims to be butter .) it could be 1 to 18...it doesn't say how many he selects.

If I understand your problem, you have a group of 18 cookies, 8 made with butter (not TO butter!) and 10 made with margarine. A man tests all 18, but simply guesses which are the butter cookies. In other words, his probability of saying "this is a butter cookie" is 1/2 for each of the 18 cookies tasted with all guesses being independent. Is that what you mean? If so, the number of "butter" guesses is binomial with parameters n = 18 and p = 0.5. If X of his guesses are "butter", some of these may be truly butter cookies and some not; the number of "butter" guesses that are truly "butter" is Y.

Is that a correct good description of your problem? If so, it is not a simple hypergeometric or a simple binomial problem. Can you see why? Can you see how conditional probabilities come into play?

How do I solve this problem with conditional probabilities ? I never done a problem like that before.
Conditional distribution
8/18= 0.4444
10/18=0.55555
Not really sure what do to with that

 

[Ray Vickson]

How do I solve this problem with conditional probabilities ? I never done a problem like that before.
Conditional distribution
8/18= 0.4444
10/18=0.55555
Not really sure what do to with that

(1) Was my description of your problem correct? Yes or no?
(2) Assuming your answer is yes, you then need to understand how you could calculate the probabilities $P(Y = k)$ for $k = 0,1,2, \ldots, 8$. You might not need to actually calculate them, but you do need to understand what you would need to do. Anyway, writing down numbers with zero explanation (like you just did) is not helpful.

So, don't start writing down numbers or random formulas yet; the time to write down numbers and (correct) formulas is after you understand the problem more thoroughly.

 

[masterchiefo]

(1) Was my description of your problem correct?es or no.
(2) Assuming your answer is yes, you then need to understand how you could calculate the probabilities $P(Y = k)$ for $k = 0,1,2, \ldots, 8$. You might not need to actually calculate them, but you do need to understand what you would need to do. Anyway, writing down numbers with zero explanation (like you just did) is not helpful.

So, don't start writing down numbers or random formulas yet; the time to write down numbers and (correct) formulas is after you understand the problem more thoroughly.

Yes your description is correct.
I am pretty confused now.
Just to make things clear do I need to use binomial or hypergeometric for this one?
and you said n = 18 and p = 0.5 , that makes q=1-0.5
what do I need to actually find ?

 

[Ray Vickson]

Yes your description is correct.
I am pretty confused now.
Just to make things clear do I need to use binomial or hypergeometric for this one?
and you said n = 18 and p = 0.5 , that makes q=1-0.5
what do I need to actually find ?

If he does not know about the 8 and the 10:

Let me repeat for one last time: in this case how would you calculate P(Y = 0)? How would you calculate P(Y = 1)? How would you calculate P(Y=2)? etc.

I will not tell you what formulas you need to use, but I will hint a little more about how you should think about the problem.

Assuming the man guesses randomly, what are the probabilities P(X = n) for n = 0,1,2, ... . Here, X = number of "butter" guesses, and could (presumably) be any number from 0 to 18 because he makes 18 random guesses.

If you know that {X = n}, what can you say about {Y = k}?

What if he knows about the 8 and the 10?
In that case he will not make more than 8 "butter" guesses, so his guesses could not be truly independent and random anymore. One simple way to deal with this would be to assume totally random and independent guesses until a total of 8 "butter"s have been reached, the declare all the remaining to be "margarine". Alternatively, if he is allowed to calculate, he could adapt his "guess" probabilities to his previous answers. For instance he could guess "butter" on his first cookie with probability 8/18. Then if he guessed "butter" first, he is left with 17 cookies among which he thinks 7 are "butter" and 10 are "marg". He could choose his second guess probability accordingly. Similarly, if he guessed "marg" first he is now looking at 17 cookies of which he thinks 8 are "butter" and 9 are "marg", etc.

 

[masterchiefo]

If he does not know about the 8 and the 10:

Let me repeat for one last time: in this case how would you calculate P(Y = 0)? How would you calculate P(Y = 1)? How would you calculate P(Y=2)? etc.

I will not tell you what formulas you need to use, but I will hint a little more about how you should think about the problem.

Assuming the man guesses randomly, what are the probabilities P(X = n) for n = 0,1,2, ... . Here, X = number of "butter" guesses, and could (presumably) be any number from 0 to 18 because he makes 18 random guesses.

If you know that {X = n}, what can you say about {Y = k}?

What if he knows about the 8 and the 10?
In that case he will not make more than 8 "butter" guesses, so his guesses could not be truly independent and random anymore. One simple way to deal with this would be to assume totally random and independent guesses until a total of 8 "butter"s have been reached, the declare all the remaining to be "margarine". Alternatively, if he is allowed to calculate, he could adapt his "guess" probabilities to his previous answers. For instance he could guess "butter" on his first cookie with probability 8/18. Then if he guessed "butter" first, he is left with 17 cookies among which he thinks 7 are "butter" and 10 are "marg". He could choose his second guess probability accordingly. Similarly, if he guessed "marg" first he is now looking at 17 cookies of which he thinks 8 are "butter" and 9 are "marg", etc.

The problem where he knows about the 8 butter is already resolved.

 

[Ray Vickson]

The problem where he knows about the 8 butter is already resolved.

No, I don't think it has been; I do not believe your solution.

Among other things, it does not deal with the issue I outlined above: exactly how does he "guess" when the 8 and 10 are known to him? You just ignored that aspect altogether, but it cannot be ignored.

 

[masterchiefo]

If he does not know about the 8 and the 10:

Let me repeat for one last time: in this case how would you calculate P(Y = 0)? How would you calculate P(Y = 1)? How would you calculate P(Y=2)? etc.

I will not tell you what formulas you need to use, but I will hint a little more about how you should think about the problem.

Assuming the man guesses randomly, what are the probabilities P(X = n) for n = 0,1,2, ... . Here, X = number of "butter" guesses, and could (presumably) be any number from 0 to 18 because he makes 18 random guesses.

If you know that {X = n}, what can you say about {Y = k}?

What if he knows about the 8 and the 10?
In that case he will not make more than 8 "butter" guesses, so his guesses could not be truly independent and random anymore. One simple way to deal with this would be to assume totally random and independent guesses until a total of 8 "butter"s have been reached, the declare all the remaining to be "margarine". Alternatively, if he is allowed to calculate, he could adapt his "guess" probabilities to his previous answers. For instance he could guess "butter" on his first cookie with probability 8/18. Then if he guessed "butter" first, he is left with 17 cookies among which he thinks 7 are "butter" and 10 are "marg". He could choose his second guess probability accordingly. Similarly, if he guessed "marg" first he is now looking at 17 cookies of which he thinks 8 are "butter" and 9 are "marg", etc.

No, I don't think it has been; I do not believe your solution.

Among other things, it does not deal with the issue I outlined above: exactly how does he "guess" when the 8 and 10 are known to him? You just ignored that aspect altogether, but it cannot be ignored.

Where he knows about the 8 butter, its resolved and corrected by the teacher himself last week.

My problem today is where he does not know about the number of butter.
Sorry, I been thinking, thinking and I still don't understand.

Before I use the correct formula for p(x).

I want to know if this is correct:
n=18
p=0.5
N1=8
N2=10
N=N1+N2=18

k=0 to 8
And I would use the Hypergeometric P(x) formula

,

 

[masterchiefo]

No, I don't think it has been; I do not believe your solution.

Among other things, it does not deal with the issue I outlined above: exactly how does he "guess" when the 8 and 10 are known to him? You just ignored that aspect altogether, but it cannot be ignored.

This is for P(Y=K) K 0 to 8
binomPdf(18,0.5,{0,1,2,3,4,5,6,7,8})
{3.814697265625E−6,6.8664550781251E−5,5.8364868164062E−4,0.00311279296875,0.011672973632812,0.032684326171874,0.070816040039061,0.12139892578125,0.16692352294921}

 

[masterchiefo]

Can someone please help me, I am on this problem since yesterday and I can't figure it out.
It is the first time I am that confused about a probability problem :/

 

[Ray Vickson]

This is for P(Y=K) K 0 to 8
binomPdf(18,0.5,{0,1,2,3,4,5,6,7,8})
{3.814697265625E−6,6.8664550781251E−5,5.8364868164062E−4,0.00311279296875,0.011672973632812,0.032684326171874,0.070816040039061,0.12139892578125,0.16692352294921}

These are the probabilities P(X = n) for n = 0, ..., 8 (that is, the probability that the man guesses n). That is not what you want: you want the probability P(Y = k) for k = 0, ..., 8, which are the probabilities that k of his guesses are correct.

There is no reason for X to stop at 8, because if the man knows nothing at all about the true numbers of butter/marg cookies he can guess any number from 0 10 18. The extreme cases X=0 means he guesses there are no butter cookies and X = 18 means he guesses that all 18 are butter cookies. Of course, unknown to him (but known to US) the actual number of butter cookies is 8 exactly.

 

[masterchiefo]

These are the probabilities P(X = n) for n = 0, ..., 8 (that is, the probability that the man guesses n). That is not what you want: you want the probability P(Y = k) for k = 0, ..., 8, which are the probabilities that k of his guesses are correct.

There is no reason for X to stop at 8, because if the man knows nothing at all about the true numbers of butter/marg cookies he can guess any number from 0 10 18. The extreme cases X=0 means he guesses there are no butter cookies and X = 18 means he guesses that all 18 are butter cookies. Of course, unknown to him (but known to US) the actual number of butter cookies is 8 exactly.

Sorry man, but I need more details about what I have to do. I am entirely clueless now about what to do.

I just calculated P(Y=k) but you said its wrong and I don't understand.
This is for P(Y=K) K 0 to 8
binomPdf(18,0.5,{0,1,2,3,4,5,6,7,8})

what is wrong with that?, or do I have to change 18 to 8?

 

[Ray Vickson]

Sorry man, but I need more details about what I have to do. I am entirely clueless now about what to do.

I just calculated P(Y=k) but you said its wrong and I don't understand.
This is for P(Y=K) K 0 to 8
binomPdf(18,0.5,{0,1,2,3,4,5,6,7,8})

what is wrong with that?, or do I have to change 18 to 8?

No, you did NOT calculate P(Y = k) for k from 0 to 8; you calculated P(X = k) for k from 0 to 8.

I have explained already what are X and Y, and will not do so again. It is against PF rules for me to tell you exactly what you need to do. I have given several hints, but that is all I am allowed to do (or am willing to do).

 

[masterchiefo]

No, you did NOT calculate P(Y = k) for k from 0 to 8; you calculated P(X = k) for k from 0 to 8.

I have explained already what are X and Y, and will not do so again. It is against PF rules for me to tell you exactly what you need to do. I have given several hints, but that is all I am allowed to do (or am willing to do).

exactly so I just do binomPdf(8,0.5,{0,1,2,3,4,5,6,7,8})
since we know there is 8 butter biscuit

 

[masterchiefo]

No, you did NOT calculate P(Y = k) for k from 0 to 8; you calculated P(X = k) for k from 0 to 8.

I have explained already what are X and Y, and will not do so again. It is against PF rules for me to tell you exactly what you need to do. I have given several hints, but that is all I am allowed to do (or am willing to do).

Also, why is P=0.5 like you said earlier ? We don't have the same amount of Butter biscuit and Margarine Biscuit.

 

[Ray Vickson]

Also, why is P=0.5 like you said earlier ? We don't have the same amount of Butter biscuit and Margarine Biscuit.

You said that the man guesses at random. He has no idea about the number of biscuit types (WE know it, but he does not). So, I just assumed he guesses both types to be equally likely each time.

Anyway, your reasoning is absent, so when you just write something down without justification that is not a correct argument. The distribution you get may or may not be correct, but you have not given any reasons why it should be! If you want to be believed you cannot just write these things down without any explanations.

 

[masterchiefo]

You said that the man guesses at random. He has no idea about the number of biscuit types (WE know it, but he does not). So, I just assumed he guesses both types to be equally likely each time.

Anyway, your reasoning is absent, so when you just write something down without justification that is not a correct argument. The distribution you get may or may not be correct, but you have not given any reasons why it should be! If you want to be believed you cannot just write these things down without any explanations.

If we know P=0.5 and n=18 so q=1-0.5, so why do I even need to calculate the binomial then?

I just need to know the variance and in Binomial the variance =n*p*q

 

[Ray Vickson]

If we know P=0.5 and n=18 so q=1-0.5, so why do I even need to calculate the binomial then?

I just need to know the variance and in Binomial the variance =n*p*q

You have not convinced me that the distribution of Y is actually binomial! If you have a proof, please present it.

That is my last word on the subject.

 

[masterchiefo]

You have not convinced me that the distribution of Y is actually binomial! If you have a proof, please present it.

That is my last word on the subject.

That is the problem in
Hypergeometric the variannce = n*p*q * ((N-n)/(N-1))
and I did this:
18*0.5*0.5 * ((18-18)/(18-1)) and this equal to 0.

which makes no sense.

 

[masterchiefo]

You have not convinced me that the distribution of Y is actually binomial! If you have a proof, please present it.

That is my last word on the subject.

and you said it yourself in the start that it is binomial with p=0.5 and n=18 and that confused the hell out of me and now you are telling me its not binomial. I have put 2 days into this problem and now you are blaming me.

 

[masterchiefo]

If so, the number of "butter" guesses is binomial with parameters n = 18 and p = 0.5.

That is your quote where you said its binomial and that alone confused me. because from the beginning I had hypergeometric in mind then for 2 days I was trying to resolve this problem with binomial.

 

[Ray Vickson]

and you said it yourself in the start that it is binomial with p=0.5 and n=18 and that confused the hell out of me and now you are telling me its not binomial. I have put 2 days into this problem and now you are blaming me.

No, I never said that; you mis-read what I wrote.

I said that X = number of guesses that say "milk" is a binomial random variable with distribution Binomial(18, 0.5). However, X is not at all what you want to know about: you want to know Y = number of actual "milks" among the X. So, for example, if X = 7 (the man said "milk" 7 times out of his 18 tastings), there could be anywhere from 0 to 7 actual "milk" cookies in that chosen 7. That actual number is what is meant by Y, and that is the random variable you are supposed to find out about---NOT X!.

 

[masterchiefo]

No, I never said that; you mis-read what I wrote.

I said that X = number of guesses that say "milk" is a binomial random variable with distribution Binomial(18, 0.5). However, X is not at all what you want to know about: you want to know Y = number of actual "milks" among the X. So, for example, if X = 7 (the man said "milk" 7 times out of his 18 tastings), there could be anywhere from 0 to 7 actual "milk" cookies in that chosen 7. That actual number is what is meant by Y, and that is the random variable you are supposed to find out about---NOT X!.

so P(y=k) is the probability and then I will use that number in my hypergeometric variance formula, is that right ?

 

[masterchiefo]

No, I never said that; you mis-read what I wrote.

I said that X = number of guesses that say "milk" is a binomial random variable with distribution Binomial(18, 0.5). However, X is not at all what you want to know about: you want to know Y = number of actual "milks" among the X. So, for example, if X = 7 (the man said "milk" 7 times out of his 18 tastings), there could be anywhere from 0 to 7 actual "milk" cookies in that chosen 7. That actual number is what is meant by Y, and that is the random variable you are supposed to find out about---NOT X!.

Sorry english is not my primary language so I tend to confuse and misread.

Okay so:
k being 0 to 8
HyperCDF(n,N1,N2,k) = 1
HyperCDF(18,8,10,0,8) = 1

Now the Variance formula for Hypergeo:
n=18
N=18
p=1
q=1-p
variance= npq*((N-n)/(N-1))
=18*1*(1-1)*((18-18)/(18-1))
= 0

Something is obviously wrong :(

 

[Ray Vickson]

so P(y=k) is the probability and then I will use that number in my hypergeometric variance formula, is that right ?

You are getting closer. The actual hypergeometric you must use depends on X. If X = 7 (the man thinks there are 7 "milks") the number of actual milks among those 7 has a hypergeometric distribution with parameters (8,10,7). If X = 3 (the thinks there are 3 "milks") you need to use hypergeometric with parameters (8,10,3), etc.

 

[masterchiefo]

You are getting closer. The actual hypergeometric you must use depends on X. If X = 7 (the man thinks there are 7 "milks") the number of actual milks among those 7 has a hypergeometric distribution with parameters (8,10,7). If X = 3 (the thinks there are 3 "milks") you need to use hypergeometric with parameters (8,10,3), etc.

What we know is that there is 8 butter. To make things actually go forward can you least say what is good and what is wrong because everytime I just try to completely rethink everything because I have no idea what is actually right and that is why its been 2 days and its not resolved.

I don't even know how many he selects, so I can't set the (n), it can't be 18 because that would mean he selected 18 that he thinks are butter, but that's not true.

HyperCDF(n,N1,N2,k) = 1
n=18
N1=8
N2=10
K=0 to 8

 

[Ray Vickson]

What we know is that there is 8 butter. To make things actually go forward can you least say what is good and what is wrong because everytime I just try to completely rethink everything because I have no idea what is actually right and that is why its been 2 days and its not resolved.

I don't even know how many he selects, so I can't set the (n), it can't be 18 because that would mean he selected 18 that he thinks are butter, but that's not true.

HyperCDF(n,N1,N2,k) = 1
n=18
N1=8
N2=10
K=0 to 8

The number (he selects is random, not known before.

What we know is that there is 8 butter. To make things actually go forward can you least say what is good and what is wrong because everytime I just try to completely rethink everything because I have no idea what is actually right and that is why its been 2 days and its not resolved.

I don't even know how many he selects, so I can't set the (n), it can't be 18 because that would mean he selected 18 that he thinks are butter, but that's not true.

HyperCDF(n,N1,N2,k) = 1
n=18
N1=8
N2=10
K=0 to 8

Of course you do not know how many he selects; that is a random variable ranging from 0 to 18. Indeed, he CAN select 18 if he guesses that all of them are milk cookies; he would be wrong, but he does not know that. It is not very probable that he does select 18; the probability would be the same as that for getting 18 heads when you toss a coin 18 times: unlikely, but definitely possible. Similarly, it is not very likely that he selects 0, but it is possible: that has a nonzero probability.

You do not know the actual number he selects, but you DO KNOW its probability distribution. That is all you need in order to determine the number of successfully-detected milk cookies Y, which is also random and can range from 0 to 8. You need to use some pretty fundamental results and methods in probability in order to finish the computation, and I really am not permitted to say much more.