Probability of picking certain container based on content: red, blue marbles, 2 urns

Bandicoot

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Hi, I'm sitting with this question that has been bugging me:

"You have two containers, A and B, each holds two types of marbles, red and blue.

Container A holds 30 red and 10 blue.
Container B holds 20 red and 20 blue.

You pick a random container and pick up one red marble.


What are the probability that you picked container A? "

Reading the question, I assume the answer is 50% - since the question reads; "You pick a container, then pick a marble", the marbles have nothing to do with what container I picked.. So it is a trick question yes?

On the other hand, had the question read, "you pick a marble, it is red, what is the probaility that you picked it from container A?" then;

Out of 80 marbles, theres is (50/80)*100 = 62,5 % chance of picking a red.
Out of the 50 red marbles, theres a (30/50)*100 = 60 % of that red being from container A.




What do you guess think? How do you read the question?



Thanks!
 
Last edited:
Hi, I'm sitting with this question that has been bugging me:


You have two containers, A and B, each holds two types of marbles, red and blue.

Container A holds 30 reb and 10 blue.
Container B holds 20 red and 20 blue.

You pick a random container and pick up one red marble.
What are the probability that you picked from container A?

Thanks!
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Reading the question, I assume the answer is 50% - since the question reads; "You pick a container, then pick a marble", the marbles have nothing to do with what container I picked.. So it is a trick question yes?

On the other hand, had the question read, "you pick a marble, it is red, what is the probaility that you picked it from container A?" then;

Out of 80 marbles, theres is (50/80)*100 = 62,5 % chance of picking a red.
Out of the 50 red marbles, theres a (30/50)*100 = 60 % of that red being from container A.
 
The question is 'Given the information that you picked a red marble, what is the probability you picked container A?" Without any information, the two containers would be equally likely but it should occur to you that, since A contains more red marbles than B, since you picked a red marble, it was more likely to have come from A than B. But how much more likely?

Imagine doing this experiment 1000 times. 500 times you pick from A and 500 times you pick from B. Of the 500 times you pick from A, you get a red marble 3/4 of the time: (3/4)(500)= 375 red marbles and 500- 375= 125 blue marbles. Of the 500 times you pick from B, you get a red marble 1/2 the time, 250 red marbles and 250 blue marbles.

So you have a total of 250+ 375= 625 red marbles, 375 from A, 250 from B. Given that the marble picked was red, the probability that it came from A was 375/625= 75/105= 15/21= 5/7.
 
Reading the question, I assume the answer is 50% - since the question reads; "You pick a container, then pick a marble", the marbles have nothing to do with what container I picked.. So it is a trick question yes?

On the other hand, had the question read, "you pick a marble, it is red, what is the probaility that you picked it from container A?" then;

Out of 80 marbles, theres is (50/80)*100 = 62,5 % chance of picking a red.
Out of the 50 red marbles, theres a (30/50)*100 = 60 % of that red being from container A.
Think about this, what if container A had no red marbles. Then the probability that the red marble came from A is 0, not .5. What if A had some red marbles and B had no red marbles. Then if you picked a red marble it is certain that you picked it from A. The only way to get 50% is if A and B both had the same percent of red marbles (and blue marbles). One more example, what if A had 99,999 red marbles and 1 blue marble and B had 22 blue marbles and 1 red. Doesn't it seem that a red marble is more likely to come from A than B?? Think about this.
 
Well yeah, and I agree with HallsofIvy, except I get 60 % chance and not 71 %. (30/50)*100 = 60 %...


However, the way the question is written makes me doubt.


"You pick a random container and pick up a red marble - what are the chances you picked container A?"

To me, this means; You already picked a container. This is an isolated event and leaves you with 50/50 chance of picking container A. The marbles are meaningless.



Had the question been; "You pick a random red marble - what the chances the marble originates from container A", then yes, 60 %, because container A has more marbles.
 
Well yeah, and I agree with HallsofIvy, except I get 60 % chance and not 71 %. (30/50)*100 = 60 %...

However, the way the question is written makes me doubt.

"You pick a random container and pick up a red marble - what are the chances you picked container A?"

To me, this means; You already picked a container. This is an isolated event and leaves you with 50/50 chance of picking container A. The marbles are meaningless.

Had the question been; "You pick a random red marble - what the chances the marble originates from container A", then yes, 60 %, because container A has more marbles.

Yes, HallsofIvy is correct, apart from an arithmetic error: rather than 375/625= 75/105= 15/21= 5/7, it is 375/625= 75/125= 15/25= 3/5. Or just do the division directly.

Traditionally, you would do essentially the same work using Bayes Theorem.

I think the question is clear, unless you twist it because it doesn't explicitly say you took the marble from the container you picked. It said,

You have two containers, A and B, each holds two types of marbles, red and blue.

Container A holds 30 red and 10 blue.
Container B holds 20 red and 20 blue.

You pick a random container and pick up one red marble [from that container].

What are [is?] the probability that you picked container A?

Is this your issue, that you don't think my addition is what is intended? I would counter that it also doesn't say that you poured out all the marbles into another container before picking one, so that all marbles would be equally likely. The natural reading is that you picked a container from which to pick a marble. Otherwise, it doesn't say how you picked the marble, and all you can say is that there is not enough information.

But since this is presumably for a math class, not an English class, I do think it should have been written more explicitly.
 
Well yeah, and I agree with HallsofIvy, except I get 60 % chance and not 71 %. (30/50)*100 = 60 %...However, the way the question is written makes me doubt."You pick a random container and pick up a red marble - what are the chances you picked container A?"To me, this means; You already picked a container. This is an isolated event and leaves you with 50/50 chance of picking container A. The marbles are meaningless.Had the question been; "You pick a random red marble - what the chances the marble originates from container A", then yes, 60 %, because container A has more marbles.
The question is not ideally worded. It means what is the probability that container A was chosen given that you picked a red ball and chose the container at random.

\(\displaystyle P(r \text { and } A) = P(r|A) * P(A) = \dfrac{30}{40} * \dfrac{1}{2} = 0.375.\)

Do you buy that?

\(\displaystyle P(r \text { and } B) = P(r|B) * P(B) = \dfrac{20}{40} * \dfrac{1}{2} = 0.25.\)

Do you buy that?

\(\displaystyle \therefore P(R) = 0.25 + 0.375 = 0.625.\)

Still with me?

\(\displaystyle P(A \text { and } r) = P(A|r) * P(r) \implies \)

\(\displaystyle P(A|r) = \dfrac{P(r \text { and } A)}{P(r)} = \dfrac{0.375}{0.625} = 0.6.\)
 
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