C
Cleopatre VII
Guest
Thank you very much for these solutions! I will now post my solutions so that you can compare them.
Solution of example 1.
a) Let us introduce the following designations:
A - a family with two boys was selected,
B - the younger child is a boy,
A∩B - the younger child is a boy and the older child is a boy.
We want to calculate the probability of event A, provided that event B happened. We will use the formula:
P(A│B)=(P(A∩B))/(P(B)).
We calculate the probability of event B. The younger child can be a boy and a girl with the same probability. Therefore:
P(B)=1/2.
We now calculate the probability of the event A∩B. The probability that the younger child is a boy is equal to 1/2 and that the older child is a boy is also equal to 1/2, hence:
P(A∩B)=1/2∙1/2=1/4.
So we finally have:
P(A│B)=(P(A∩B))/(P(B))=1/4∙2/1=2/4=1/2.
b) Let us introduce the following designations:
A - a family with two boys was selected,
B - there is at least one boy in the family,
A∩B - there are two boys in the family.
We want to calculate the probability of event A, provided that event B happened. We will use the following formula again:
P(A│B)=(P(A∩B))/(P(B)).
We calculate the probability of event B. We have two children in the family and we want at least one boy, so we have 3 options:
(boy, girl), (girl, boy), (boy, boy)
The total of all possibilities is 4:
(boy, girl), (girl, boy), (boy, boy), (girl, girl),
Therefore:
P(B)=3/4.
We now calculate the probability of the event A∩B. The probability that the younger child is a boy is equal to 1/2 and that the older child is a boy is also equal to 1/2, so:
P(A∩B)=1/2∙1/2=1/4.
So we finally have:
P(A│B)=(P(A∩B))/(P(B))=1/4∙4/3=1/3.
Solution of example 2.
We introduce the following designations:
A - an event involving the drawing of a white ball
B1 - an event involving drawing an A urn
B2 - the event consisting in drawing the urn B
We have to calculate the probability P (B1|A). We will use the Bayes formula:
P(B1│A)=(P(A|B1 )P(B1))/(P(A)).
We always use the formula for the total probability for the denominator:
P(A)=P(A│B1 )∙P(B1 )+P(A│B2 )∙P(B2 ).
We calculate the probabilities:
P(B1) = 1/2,
P(A|B1) = 9/15,
P(B2) = 1/2,
P(A|B2) = 15/20.
We substitute them into the formula:
P(A)=9/15∙1/2+15/20∙1/2=3/10+15/40=(12+15)/40=27/40.
We insert the above result into the Bayes formula:
P(B1│A)=(P(A|B1 )P(B1))/(P(A))=(9/15∙1/2)/(27/40)=(3/10)/(27/40)=4/9.
And this is our final result.
Solution of example 1.
a) Let us introduce the following designations:
A - a family with two boys was selected,
B - the younger child is a boy,
A∩B - the younger child is a boy and the older child is a boy.
We want to calculate the probability of event A, provided that event B happened. We will use the formula:
P(A│B)=(P(A∩B))/(P(B)).
We calculate the probability of event B. The younger child can be a boy and a girl with the same probability. Therefore:
P(B)=1/2.
We now calculate the probability of the event A∩B. The probability that the younger child is a boy is equal to 1/2 and that the older child is a boy is also equal to 1/2, hence:
P(A∩B)=1/2∙1/2=1/4.
So we finally have:
P(A│B)=(P(A∩B))/(P(B))=1/4∙2/1=2/4=1/2.
b) Let us introduce the following designations:
A - a family with two boys was selected,
B - there is at least one boy in the family,
A∩B - there are two boys in the family.
We want to calculate the probability of event A, provided that event B happened. We will use the following formula again:
P(A│B)=(P(A∩B))/(P(B)).
We calculate the probability of event B. We have two children in the family and we want at least one boy, so we have 3 options:
(boy, girl), (girl, boy), (boy, boy)
The total of all possibilities is 4:
(boy, girl), (girl, boy), (boy, boy), (girl, girl),
Therefore:
P(B)=3/4.
We now calculate the probability of the event A∩B. The probability that the younger child is a boy is equal to 1/2 and that the older child is a boy is also equal to 1/2, so:
P(A∩B)=1/2∙1/2=1/4.
So we finally have:
P(A│B)=(P(A∩B))/(P(B))=1/4∙4/3=1/3.
Solution of example 2.
We introduce the following designations:
A - an event involving the drawing of a white ball
B1 - an event involving drawing an A urn
B2 - the event consisting in drawing the urn B
We have to calculate the probability P (B1|A). We will use the Bayes formula:
P(B1│A)=(P(A|B1 )P(B1))/(P(A)).
We always use the formula for the total probability for the denominator:
P(A)=P(A│B1 )∙P(B1 )+P(A│B2 )∙P(B2 ).
We calculate the probabilities:
P(B1) = 1/2,
P(A|B1) = 9/15,
P(B2) = 1/2,
P(A|B2) = 15/20.
We substitute them into the formula:
P(A)=9/15∙1/2+15/20∙1/2=3/10+15/40=(12+15)/40=27/40.
We insert the above result into the Bayes formula:
P(B1│A)=(P(A|B1 )P(B1))/(P(A))=(9/15∙1/2)/(27/40)=(3/10)/(27/40)=4/9.
And this is our final result.
John G
The Living Force
I used the formulas for the 2nd one and still botched it. I started out using the 1/2 probabilities for the urns and then abandoned that since it made more than 2 events; apparently one did need to worry about more than 2 events. I via a wrong use of a formula ended up with Zar's 9/24. The first one I got right given I didn't realize it was two questions instead of redundant information.
dennis
Dagobah Resident
The way I read the exercise was that the conditions stated established that the first child is male, therefore the following possibilities in red should not be included in the calculation. Your statement "that the older child is a boy is also equal to 1/2" was understood to be a condition precedent to any further computation, so I skipped that step, or assumed that it was equal to 1.0
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C
Cleopatre VII
Guest
The philosophy of hermeticism describes the verity of consequences of the seven fundamental laws. Time is conjured in any implication relationship. Every real influence is in fact timeless. Cognition depends on broadness, not on time. “God” understands the past and future of the Universe as all phenomena happen only in accordance with the seven, ruthless laws, which in fact are one law. They constitute an ancient substance evincing in many projects called the philosopher’s stone. The source substance is a final formula, a thing that people call “God”.
To separate living matter from the inanimate matter, there exist attempts to define life in terms taken from the theory of systems. The system is defined as a specifically ordered set of elements and parts interacting with each other, forming a whole. The systems are divided into closed and open.
Living organisms, unlike inanimate matter, are the open systems, relatively isolated, constantly exchanging matter and energy with the environment, which opposes the growth of entropy and keeps its structure at a particular level. By using the terms of thermodynamics, the body is defined as a thermodynamic system which, at the expense of increasing entropy in the environment, is capable of reducing its internal entropy.
However, the concept of a system does not necessarily work in biological sciences. I wrote about stars that are not considered as living organisms, although in the system definitions they fulfill all the necessary properties. Systematization is a great tool, which is an important aid, e.g. in systems biology or genetics, but some of the concepts seem to be in essence much more philosophical and chaotic than ordered according to human understanding.
Back in the past, when I was still a high school student, I tried to somehow systematize this world. From today’s perspective, that might seem a bit naive and silly. But what I like about it is that it was very authentic at the time. There is nothing more beautiful than authenticity. Therefore, I will mention it.
In this context, the conception of the seven postulates may turn out very interesting.
The mentioned conception is consisted of the seven postulates.
The Postulate of Matter,
The Postulate of Mass,
The Postulate of Time,
The Postulate of Space,
The Postulate of Equality between Quantization and Continuation,
The Postulate of Unitary Timespace,
The Universal Postulate of Unity between Time and Consciousness.
Only the first postulate will be analized, because is strictly associated with the considered subject.
The postulate of matter claims that the structure of matter is not different than the structure of consciousness and that there occurs bijection (isomorphism) between these beings. According to this postulate, only the consciousness may observe matter, but the observation becomes impossible without matter itself. That is why matter and consciousness are associated. The consciousness of matter is, however, the awareness of the passage of time, location in space, ageing of materials, etc. Therefore, it is not worthwhile to limit to biological definitions, when considering ageing.
Moreover, a particular physical theory, which assumes consciousness as one of the states of matter may also be relevant.
In 2014 cosmologist and theoretical physicist Max Tegmark proposed the existence of a new state of matter - "perceptronium" - in which atoms are adapted to the processing of information and give rise to subjectivity and, ultimately, to consciousness ([1401.1219] Consciousness as a State of Matter).
Tegmark believes that consciousness is not in our brain or in another part of our body but can be interpreted as a mathematical formula and is the result of a specific set of mathematical conditions. In his view, different types of consciousness can arise, as are the various states of matter that arise in different conditions.
The scientist based his theory on the work of neuroscientist Giulio Tononi, who in 2008 recognized that if we want to prove that something is aware, we have to demonstrate
two features. First, the conscious being must be able to store, process and recall large amounts of information. Secondly, this information must be integrated into one unbroken whole.
Tononi integrated information theory foresees that devices as simple as a thermostat or a photoelectric diode may have some flashes of consciousness, although that does not mean that the system is aware. Neurobiologist even proposed a unit Φ (phi) to measure the consciousness of a given being - whether it would be a thermostat or a robot.
Tegmark enriched his concept and established two states of matter - "computronium", which fulfills the requirements of the first characteristic of consciousness, i.e. collects, processes, and recalls large amounts of information; and "perceptronium", which further integrates the information into an unbreakable whole. In his article he defined five basic principles that we could use to distinguish conscious matter from solids, liquids, and gases - these are the principles of information, integration, independence, dynamics, and utility.
If the presented theory is true, then most likely all complex systems have some form of consciousness. A human would have to recognize that he is not the only conscious being...
Maybe I'll say a little more about this in the next post.
To separate living matter from the inanimate matter, there exist attempts to define life in terms taken from the theory of systems. The system is defined as a specifically ordered set of elements and parts interacting with each other, forming a whole. The systems are divided into closed and open.
Living organisms, unlike inanimate matter, are the open systems, relatively isolated, constantly exchanging matter and energy with the environment, which opposes the growth of entropy and keeps its structure at a particular level. By using the terms of thermodynamics, the body is defined as a thermodynamic system which, at the expense of increasing entropy in the environment, is capable of reducing its internal entropy.
However, the concept of a system does not necessarily work in biological sciences. I wrote about stars that are not considered as living organisms, although in the system definitions they fulfill all the necessary properties. Systematization is a great tool, which is an important aid, e.g. in systems biology or genetics, but some of the concepts seem to be in essence much more philosophical and chaotic than ordered according to human understanding.
Back in the past, when I was still a high school student, I tried to somehow systematize this world. From today’s perspective, that might seem a bit naive and silly. But what I like about it is that it was very authentic at the time. There is nothing more beautiful than authenticity. Therefore, I will mention it.
In this context, the conception of the seven postulates may turn out very interesting.
The mentioned conception is consisted of the seven postulates.
The Postulate of Matter,
The Postulate of Mass,
The Postulate of Time,
The Postulate of Space,
The Postulate of Equality between Quantization and Continuation,
The Postulate of Unitary Timespace,
The Universal Postulate of Unity between Time and Consciousness.
Only the first postulate will be analized, because is strictly associated with the considered subject.
The postulate of matter claims that the structure of matter is not different than the structure of consciousness and that there occurs bijection (isomorphism) between these beings. According to this postulate, only the consciousness may observe matter, but the observation becomes impossible without matter itself. That is why matter and consciousness are associated. The consciousness of matter is, however, the awareness of the passage of time, location in space, ageing of materials, etc. Therefore, it is not worthwhile to limit to biological definitions, when considering ageing.
Moreover, a particular physical theory, which assumes consciousness as one of the states of matter may also be relevant.
In 2014 cosmologist and theoretical physicist Max Tegmark proposed the existence of a new state of matter - "perceptronium" - in which atoms are adapted to the processing of information and give rise to subjectivity and, ultimately, to consciousness ([1401.1219] Consciousness as a State of Matter).
Tegmark believes that consciousness is not in our brain or in another part of our body but can be interpreted as a mathematical formula and is the result of a specific set of mathematical conditions. In his view, different types of consciousness can arise, as are the various states of matter that arise in different conditions.
The scientist based his theory on the work of neuroscientist Giulio Tononi, who in 2008 recognized that if we want to prove that something is aware, we have to demonstrate
two features. First, the conscious being must be able to store, process and recall large amounts of information. Secondly, this information must be integrated into one unbroken whole.
Tononi integrated information theory foresees that devices as simple as a thermostat or a photoelectric diode may have some flashes of consciousness, although that does not mean that the system is aware. Neurobiologist even proposed a unit Φ (phi) to measure the consciousness of a given being - whether it would be a thermostat or a robot.
Tegmark enriched his concept and established two states of matter - "computronium", which fulfills the requirements of the first characteristic of consciousness, i.e. collects, processes, and recalls large amounts of information; and "perceptronium", which further integrates the information into an unbreakable whole. In his article he defined five basic principles that we could use to distinguish conscious matter from solids, liquids, and gases - these are the principles of information, integration, independence, dynamics, and utility.
If the presented theory is true, then most likely all complex systems have some form of consciousness. A human would have to recognize that he is not the only conscious being...
Maybe I'll say a little more about this in the next post.
John G
The Living Force
I like that Tegmark relates Tononi to quantum systems and error correction codes.
C
Cleopatre VII
Guest
I like it too. What do you think about his approach in general? Have you read the article I linked to?
Yet both Tegmark and Tononi look to me as pure materialists. The other extreme would be, of course, fundamentalists. Of these two I do not really know which one is better. Why are scientists so afraid of using the word "soul"? Penfield in his "The Mystery of the Mind" wrote:
"Direct communication between the mind of man and the mind of God is quite another matter. The argument, in favor of this, lies in the claim, made by so many men for so long a time that they have received guidance and revelation from some power beyond themselves through the medium of prayer. I see no reason to doubt this evdence, nor any means of submitting it to scientific proof.
Indeed, no scientist, by virtue of his science, has the right to pass judgment on the faiths by which men live and die. We can only set out the data about the brain, and present the physiological hypotheses that are relevant to what the mind does. "
Why is this conviction that "...no scientist, by virtue of his science, has the right to pass judgment on the faiths by which men live and die".
I really do not understand. While it s true that contemporary science is too primitive to address such questions, why are we afraid to think that science can grow into something new and better? Perhaps something like mathematical theology (that would include some abstract and general information theory) could be a good start?
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John G
The Living Force
Yes that article is the first I've seen of Tegmark talking about Tononi. I've seen things from both of them before. For me, quantum systems and error correcting codes kind of are the generalized place where I want to be since they get information theory closer to algebra:
CLASSICAL and QUANTUM INFORMATION THEORY
www.valdostamuseum.com
dennis
Dagobah Resident
Haven't the C's stated that humans are like transducers of cosmic information? Below is a simple diagram and explanation on the linked page. Plants that convert sunlight to sugars is another example. If this is a valid model for the human function, it could be extrapolated to a more complex system.
A transducer is an electronic device that converts energy from one form to another. The process of converting energy from one form to another is known as transduction.
A transducer consists of the following two important parts:
- Sensing element
- Transduction element
C
Cleopatre VII
Guest
I plan to write a little more on gravitoelectromagnetism tomorrow. Today I was reading a book "Causality, Electromagnetic Induction, and Gravitation. A Different Approach to the Theory of Electromagnetic and Gravitational Fields" by Oleg Jefimenko that Ark recommended to me and some ideas came to my mind. I will try to describe these concepts as philosophically as possible and will rather avoid using mathematical formulas unless they are sometimes necessary. I believe these issues are also related to information theory, but I will be moving towards that relatively slowly.
John G
The Living Force
Yeah the guy I quoted from earlier thought of the Urim and Tumim as some kind of quantum to classical transducer letting you use both quantum and classical error correcting code.
Indeed. Yet the devil (or "the good God", as Flaubert put it) is in the details. We know it happens, no doubt, but how? That we do not know. And we need this understanding. Through physics, mathematics, biology, chemistry, philosophy, everything that our inquiring rational minds are able to produce. There not so many scientists with pure minds that are interested in addressing this issue without prejudices. Fortunately this thread on the Forum is a step in this direction. A healthy mix of math, physics, theology, philosophy, and whatever else will prove to be necessary.
