Calculating Probability Using Bayes' Formula: Solving for P(urn III | silver)

In summary, the conversation is discussing the calculation of probability using Bayes formula and a tree. The question is asking to determine the probability of urn III being selected given that a silver coin was drawn from the urn. The tree provided does not add up to 1 and the correct calculation for this probability is P(urn III | silver) = 2/3.
  • #1
stunner5000pt
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Homework Statement
There are three urn contains coins
Urn 1: 5 gold coins
Urn 2: 3 gold, 3 silver
Urn 3: 3 silver

An urn is selected at randonm & a coin is drawn from the urn. If the selected coin is silver, what is the probability that urn III was selected

Determine the probability using Bayes formula & a tree
Relevant Equations
Bayes Formula
From my understanding of Bayes formula, it should look like something like this

[tex] P(Silver| III) = \frac{P(III | silver) \times P(silver)}{P(III)} [/tex]

now we know that P(urn III) = 1/3
and the probability of P(silver) = Pr(silver|urn I) + P(silver|urn II) + P(silver|urn III) = 1/3 (0) + 1/3 (1/2) + 1/3 (1) = 1/2

But how do i calculate P(urn 3|silver) ? Would it simply be 1/3?

If I used this, then
P(Silver| III) = (1/3)(1/2) / (1/3) = 1/2. is this correct?Using the tree, I have attached what I believe is the right tree.
Would the answer then be P(silver| urn III) = 1/3 / (1/2) = 2/3 ?
 

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  • #2
It's not clear to me whether you are trying to calculate the probability of urn 3 given silver;or, silver given urn 3.

The probability tree is the way to go, IMO. But you need to combine probabilities as you go along. The last column in the tree should sum to 1, not 2 ( as you have).
 
  • #3
PeroK said:
It's not clear to me whether you are trying to calculate the probability of urn 3 given silver;or, silver given urn 3.

The probability tree is the way to go, IMO. But you need to combine probabilities as you go along. The last column in the tree should sum to 1, not 2 ( as you have).
I believe that my usage of the Bayes formula (A|B) might be throwing you off.

The question statement is " An urn is selected at random & a coin is drawn from the urn. If the selected coin is silver, what is the probability that urn III was selected"

if using the formula, would it be the 1/2 that I got?

In the tree, doesn't it all add to 1? That is the
top probability is for urn 1 is 1/3, \
the middle two for urn 2 are 1/6 each and if you add them together you get 1/3,
and the bottom is 1/3

so if we add the 3(1/3) = 1?
 
  • #4
stunner5000pt said:
I believe that my usage of the Bayes formula (A|B) might be throwing you off.

The question statement is " An urn is selected at random & a coin is drawn from the urn. If the selected coin is silver, what is the probability that urn III was selected"

if using the formula, would it be the 1/2 that I got?
I don't think so.
stunner5000pt said:
In the tree, doesn't it all add to 1?
it doesn't. The last column adds up to 2.
stunner5000pt said:
That is the
top probability is for urn 1 is 1/3, \
the middle two for urn 2 are 1/6 each and if you add them together you get 1/3,
and the bottom is 1/3

so if we add the 3(1/3) = 1?
Those numbers are missing from your tree.

In any case, how do you interpret your tree?
 
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  • #5
stunner5000pt said:
Homework Statement:: There are three urn contains coins
Urn 1: 5 gold coins
Urn 2: 3 gold, 3 silver
Urn 3: 3 silver

An urn is selected at randonm & a coin is drawn from the urn. If the selected coin is silver, what is the probability that urn III was selected

Determine the probability using Bayes formula & a tree
Relevant Equations:: Bayes Formula

From my understanding of Bayes formula, it should look like something like this

[tex] P(Silver| III) = \frac{P(III | silver) \times P(silver)}{P(III)} [/tex]

now we know that P(urn III) = 1/3
Yes.
stunner5000pt said:
and the probability of P(silver) = Pr(silver|urn I) + P(silver|urn II) + P(silver|urn III) = 1/3 (0) + 1/3 (1/2) + 1/3 (1) = 1/2
You mean P(silver) = P(silver|urn I)P(urn I) + P(silver|urn II)P(urn II) + P(silver|urn III)P(urn III), which is what you calculated.
EDIT to match his calculations: P(silver) = P(urn I)P(silver|urn I) + P(urn II)P(silver|urn II) + P(urn III)P(silver|urn III)
stunner5000pt said:
But how do i calculate P(urn 3|silver) ? Would it simply be 1/3?
It is whatever you get when you plug the numbers into the Relevant Equation you gave us.
Rearrange the Relevant Equation to get P(urn III | silver) = P(silver | urn III)P(urn III) / P(silver).
stunner5000pt said:
If I used this, then
P(Silver| III) = (1/3)(1/2) / (1/3) = 1/2. is this correct?
No. P(urn III | silver) = P(silver | urn III)P(urn III) / P(silver) = ???.
stunner5000pt said:
Using the tree, I have attached what I believe is the right tree.
Would the answer then be P(silver| urn III) = 1/3 / (1/2) = 2/3 ?
Yes,
CORRECTION: I think you mean P(urn III | silver = 2/3. That is what the problem asked for.
 
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  • #6
stunner5000pt said:
Using the tree, I have attached what I believe is the right tree.
Would the answer then be P(silver| urn III) = 1/3 / (1/2) = 2/3 ?
FactChecker said:
Yes.
Except ##P(silver| III) =1##, as that is the probability of silver given urn III.
 
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  • #7
PeroK said:
Except ##P(silver| III) =1##, as that is the probability of silver given urn III.
Thanks! I stand corrected. I'm sure that he meant P( urn III | silver), which is what the problem asked for. I corrected my post.
 
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Related to Calculating Probability Using Bayes' Formula: Solving for P(urn III | silver)

What is Bayes' formula and how does it relate to probability?

Bayes' formula, also known as Bayes' theorem, is a mathematical formula used to calculate the probability of an event occurring based on prior knowledge or information. It is named after the 18th century English mathematician Thomas Bayes. It is used to update the probability of an event as more evidence or information becomes available.

What are the components of Bayes' formula?

Bayes' formula consists of two main components: the prior probability and the likelihood. The prior probability is the initial probability of an event occurring based on previous knowledge or experience. The likelihood is the probability of observing the evidence or data if the event is true. These two components are multiplied together and divided by the sum of all possible outcomes to calculate the posterior probability.

How is Bayes' formula used in real-world applications?

Bayes' formula has many applications in various fields such as medicine, finance, and machine learning. In medicine, it is used to calculate the probability of a patient having a disease based on their symptoms and other factors. In finance, it is used to update the probability of a stock's performance based on new market information. In machine learning, it is used to update the probability of a hypothesis being true based on new data.

What are the limitations of Bayes' formula?

One limitation of Bayes' formula is that it assumes that the prior probability and likelihood are independent of each other. This may not always be the case in real-world scenarios. Additionally, it requires accurate prior knowledge or data to produce reliable results. If the prior probability is incorrect, the posterior probability will also be incorrect.

How does Bayes' formula differ from other probability theories?

Bayes' formula differs from other probability theories, such as classical and frequentist probability, in that it takes into account prior knowledge or information. It also allows for the probability to be updated as new evidence or data becomes available. Other theories may not consider prior knowledge or only use the current data to calculate probability.

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