# How likely are the following deviations to the Law of Large Numbers?

#### JoelB

##### New member
This is extremely important for a major paper I am working on. I need answers for multiple questions -

1. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 798 correct guesses after playing 4593 lotteries when I should only have guessed 574.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

2. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 303 correct guesses after playing 1358 lotteries when I should only have guessed 169.75 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

3. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 84 correct guesses after playing 297 lotteries when I should only have guessed 37.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 different 6 ball combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

5. If I throw a ten sided dice (each side numbered 1 to 10) 80 times and the number 7 (just for example. Could be any number from 1 to 10) comes up a total of 15 times instead of 8 times (the expected value), is that still within the standard deviation for 80 trials? If not, how many standard deviations is that?

Thank you very much for your time.
4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? cdscf4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

#### Subhotosh Khan

##### Super Moderator
Staff member
This is extremely important for a major paper I am working on. I need answers for multiple questions -

1. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 798 correct guesses after playing 4593 lotteries when I should only have guessed 574.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

2. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 303 correct guesses after playing 1358 lotteries when I should only have guessed 169.75 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

3. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 84 correct guesses after playing 297 lotteries when I should only have guessed 37.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 different 6 ball combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

5. If I throw a ten sided dice (each side numbered 1 to 10) 80 times and the number 7 (just for example. Could be any number from 1 to 10) comes up a total of 15 times instead of 8 times (the expected value), is that still within the standard deviation for 80 trials? If not, how many standard deviations is that?

Thank you very much for your time.
4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? cdscf4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?
What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

http://www.freemathhelp.com/for

#### Jomo

##### Elite Member
This is extremely important for a major paper I am working on. I need answers for multiple questions -

1. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 798 correct guesses after playing 4593 lotteries when I should only have guessed 574.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

2. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 303 correct guesses after playing 1358 lotteries when I should only have guessed 169.75 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

3. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 84 correct guesses after playing 297 lotteries when I should only have guessed 37.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 different 6 ball combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

5. If I throw a ten sided dice (each side numbered 1 to 10) 80 times and the number 7 (just for example. Could be any number from 1 to 10) comes up a total of 15 times instead of 8 times (the expected value), is that still within the standard deviation for 80 trials? If not, how many standard deviations is that?

Thank you very much for your time.
4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? 4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely? cdscf4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?
Hi, what have you tried? Where are you stuck? We can't help you unless we see your work, even if you know it is wrong. That way we can guide you to the correct result. Please post back.

#### JoelB

##### New member
I see this is a forum where answers are not given unless you are having a problem. I have no idea how to work out the probabilities. I came for answers after having a problem posting these questions on Yahoo Answers. I see that I will have to post my questions there, since it looks like I am not going to be getting any answers here.

#### Dr.Peterson

##### Elite Member
I see this is a forum where answers are not given unless you are having a problem. I have no idea how to work out the probabilities. I came for answers after having a problem posting these questions on Yahoo Answers. I see that I will have to post my questions there, since it looks like I am not going to be getting any answers here.
You certainly will get answers here, if you ask for specific help. Just posting the problem doesn't tell us what help you need, and many of us don't like wasting time telling someone more than he needs to know about a long list of questions. Our main goal is to help people learn to solve problems themselves, not just to be free consultants (though some may not mind doing that).

I think you are saying that you know nothing at all about the calculations, and perhaps don't want to, but just want answers to include in your paper. Perhaps if you tell us what you do know, and how these fit into the paper (in particular, whether you need to be able to show where the answers come from, and to be able to defend your work), we will be better able to help. Let me take a look at the first.

1. How many standard deviations from the mean would it be if in a lottery where there are 8 balls in a jar and I only have to guess which 1 ball will come up, that I correctly guess 798 correct guesses after playing 4593 lotteries when I should only have guessed 574.12 correct guesses and how likely are the odds of this happening - extremely unlikely or totally impossible?

As I understand this, you understand that on average 1/8 of your guesses should be right: 4593/8 = 574.125. You don't mention knowing that this is a binomial distribution question, though you clearly do know something about standard deviations. What we have here is that n=4593, p=1/8, so the mean is np = 4593*1/8 = 574.125 and the standard deviation is sqrt[np(1 - p)] = sqrt[4593*.125*.875] = sqrt(502.359375) = 22.4. The number of standard deviations from the mean is z = (x - mean)/s.d. = (798 -
574.125)/22.4 = 9.99. That's essentially 10 standard deviations, which is very high.

We can never call anything in probability "totally impossible", but it is very highly unlikely. The appropriate probability is not what is given by the formula, P(X = x) = (n! / [x! (n - x)!]) * px * (1 - p)n - x. What you really need is P(X >= x), which can be found by the normal approximation, or by technology (there are various binomial calculators on the web, if you don't have anything else). This calculator says it is < 0.000001.

Is that enough to at least get you started?

#### JoelB

##### New member
@ Dr Peterson

Thank you so much! That was fantastic and just the results I were looking for. I realize this is a forum to get help and not to get answers but could you be so kind as to work out the probabilities for questions 2 to 5? This is very important work and I am only here because of a glitch on Yahoo answers.

Joel.

#### JoelB

##### New member
@Dr Peterson

I just calculated the probabilities of questions 1,2,3 and 5 using the calculator you linked. So could be be so kind as to calculate the probability of question 4 regarding a 6 out of 37 ball lottery?

Thank you so much for your time.

Joel.

#### Dr.Peterson

##### Elite Member
@Dr Peterson

I just calculated the probabilities of questions 1,2,3 and 5 using the calculator you linked. So could be be so kind as to calculate the probability of question 4 regarding a 6 out of 37 ball lottery?

Thank you so much for your time.

Joel.
I'm not even quite sure what it means:

4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 different 6 ball combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

How are combinations "sent"? What is "that draw"? I think maybe you are participating in 6 drawings, and you win (at least?) 4 of those times by matching (at least?) 5 of the numbers.

Also, you haven't given any context: How do these question relate to your "paper" -- is the "paper" a test consisting of these questions and you are asking us to do it for you, or is the "paper" some sort of research paper to which these answers contribute in some way I can't imagine? This one in particular is rather complicated, suggesting that you are expected to know something about probability! Why are you asking?

#### JoelB

##### New member
I'm not even quite sure what it means:
4. In a 6 out of 37 ball lottery where 1,777 combinations of 6 are sent for 1 draw, how likely is it that a total of 4 different 6 ball combinations out of the 1,777 different 6 ball combinations sent will match 5 out of the 6 balls drawn in that draw - likely, not likely or extremely unlikely?

How are combinations "sent"? What is "that draw"? I think maybe you are participating in 6 drawings, and you win (at least?) 4 of those times by matching (at least?) 5 of the numbers.

Also, you haven't given any context: How do these question relate to your "paper" -- is the "paper" a test consisting of these questions and you are asking us to do it for you, or is the "paper" some sort of research paper to which these answers contribute in some way I can't imagine? This one in particular is rather complicated, suggesting that you are expected to know something about probability! Why are you asking?
@ Dr Peterson

Actually this is part of a presentation I am preparing for a group.

The question regards only one lottery draw where 1,777 combinations (example of one combination - 2,8,15,27,30,35) participate in that one draw. In lotto draws one can match 3 out of the 6 balls drawn, 4 out of the 6 balls drawn, 5 out of the 6 balls drawn and / or match all 6 out of the 6 balls drawn. So my question is what is the probability that with 1,777 different combinations, 4 of them match 5 out of the 6 balls drawn?

Thanks again.

#### Dr.Peterson

##### Elite Member
@ Dr Peterson

Actually this is part of a presentation I am preparing for a group.

The question regards only one lottery draw where 1,777 combinations (example of one combination - 2,8,15,27,30,35) participate in that one draw. In lotto draws one can match 3 out of the 6 balls drawn, 4 out of the 6 balls drawn, 5 out of the 6 balls drawn and / or match all 6 out of the 6 balls drawn. So my question is what is the probability that with 1,777 different combinations, 4 of them match 5 out of the 6 balls drawn?

Thanks again.
I still don't understand. Where does the number 1777 come from, and what does it mean for a combination to "participate"? I can only picture that meaning that there have been 1777 random drawings, and you are asking for the probability that you won 4 of them by matching 5 balls with your choices. Or, equivalently, 1777 people have picked random sets of numbers, and you want the probability that 4 of them won. Maybe that's what you mean.

And, again, in talking about this sort of probability, we usually ask about the probability of winning at least 4 times, and matching at least 5 balls, not limiting it to the specific case in question. But without a context as to why you are asking, I can't be sure what is the right question to ask. Probability can be picky!

#### Dr.Peterson

##### Elite Member
Or, equivalently, 1777 people have picked random sets of numbers, and you want the probability that 4 of them won. Maybe that's what you mean.

And, again, in talking about this sort of probability, we usually ask about the probability of winning at least 4 times, and matching at least 5 balls, not limiting it to the specific case in question.
I'll suppose that we want the probability that at least 4 of the 1777 tickets win, where winning means at least 5 numbers match.

First, the probability of a ticket winning is the probability that either 5 or 6 numbers match. There are 37C6 = 2,324,784 possible tickets, of which 1 matches all 6 and 6*32 match 5 numbers (6 ways to choose which number doesn't match, and 37-5 = 32 ways to choose what it is). This gives p = (1 + 192)/2,324,784 = 8.3e-5 = 0.000083.

Now use this in the calculator, with n = 1777 and x = 4.

#### JoelB

##### New member
I'll suppose that we want the probability that at least 4 of the 1777 tickets win, where winning means at least 5 numbers match.

First, the probability of a ticket winning is the probability that either 5 or 6 numbers match. There are 37C6 = 2,324,784 possible tickets, of which 1 matches all 6 and 6*32 match 5 numbers (6 ways to choose which number doesn't match, and 37-5 = 32 ways to choose what it is). This gives p = (1 + 192)/2,324,784 = 8.3e-5 = 0.000083.

Now use this in the calculator, with n = 1777 and x = 4.
Brilliant! Just the kind of probability I was expecting. Thank you so much for your time and have a great day!

Joel.