Probability of draw with replacement

[Tajeshwar]

Homework Statement

An urn contains 50 marbles – 40 blue and 10 white. After 50 draws, exactly 40 blue and 10 white are observed.

You are not told whether the draw was done “with replacement” or “without replacement.”

What is the probability that the draw was done with replacement?

Homework Equations

The Attempt at a Solution

[/B]
I am actually not sure where to start on this one. The only thing I can think about is that if the marbles are being replaced the trials would be independent.

Sorry I don't have a better attempt for this one.

 

[BvU]

Sorry I don't have a better attempt for this one

Not good enough according to the guidelines

Least you can do is provide the relevant equations to calculate the probability of the outcome in the two scenarios. And then attempt something with those

 

[PeroK]

Homework Statement

An urn contains 50 marbles – 40 blue and 10 white. After 50 draws, exactly 40 blue and 10 white are observed.

You are not told whether the draw was done “with replacement” or “without replacement.”

What is the probability that the draw was done with replacement?

Homework Equations

The Attempt at a Solution

[/B]
I am actually not sure where to start on this one. The only thing I can think about is that if the marbles are being replaced the trials would be independent.

Sorry I don't have a better attempt for this one.

I don't like the question the way it is. In fact, I would claim there is no definite answer. So, I would change it. This may make it even harder to understand or it may give you a key to solving it. My change is:

Your friend has the said jar. They toss a coin. If it's heads, they take 50 balls without replacement; if it's tails, they take 50 balls with replacement. The result is 40 blue and 10 white. What is the probability the coin was tails?

Hint: probability tree.

 

[BvU]

50% if it's an honest coin
Did I break any PF rules stating this ?

 

[PeroK]

50% if it's an honest coin
Did I break any PF rules stating this ?

I didn't have you down as a probability denier!

 

[SammyS]

...
Did I break any PF rules stating this ?

Probably not.

 

[Tajeshwar]

Thank you for your responses.

The first problem I struggled with was understanding P(getting this combination|no replacement) = 1.

After some thought and discussion I figured this out because nothing will be left in the container.

Next I was having trouble with getting some of the values for the bayes theorem formula. I forgot using the law of indifference for choosing one or the other so we can get .5 for one or the other.

Then I used the binomial theorem to get the forward probability part of the problem. 50 C 40 (40/50)^40 (10/50)^10.

Then I used the Bayes theorem with this answer and the .5s from the indifference principle to get P(with replacement|we get this combination 40 blue and 10 white)...

I just got overwhelmed initially because I didn't realize the P(getting combo|without replacement) = 1 and the .5 probability through indifference.

 

[Tajeshwar]

I don't like the question the way it is. In fact, I would claim there is no definite answer. So, I would change it. This may make it even harder to understand or it may give you a key to solving it. My change is:

Your friend has the said jar. They toss a coin. If it's heads, they take 50 balls without replacement; if it's tails, they take 50 balls with replacement. The result is 40 blue and 10 white. What is the probability the coin was tails?

Hint: probability tree.

Yes thank you. This was a big part of what was confusing me. The .5 prob of picking either with replacement or without.

 

[Ray Vickson]

Thank you for your responses.

The first problem I struggled with was understanding P(getting this combination|no replacement) = 1.

After some thought and discussion I figured this out because nothing will be left in the container.

Next I was having trouble with getting some of the values for the bayes theorem formula. I forgot using the law of indifference for choosing one or the other so we can get .5 for one or the other.

Then I used the binomial theorem to get the forward probability part of the problem. 50 C 40 (40/50)^40 (10/50)^10.

Then I used the Bayes theorem with this answer and the .5s from the indifference principle to get P(with replacement|we get this combination 40 blue and 10 white)...

I just got overwhelmed initially because I didn't realize the P(getting combo|without replacement) = 1 and the .5 probability through indifference.

I think perhaps the problem was very badly worded.

You wrote "After 50 draws, exactly 40 blue and 10 white are observed." That seems to say that you look into the urn after all the draws have been made, and you see 40 blue and 10 white; that is what the word "after" says to me. Of course, in that interpretation, if you see anything at all in the urn after 50 draws, all those draws must have been made with replacement; so the (conditional) probability sought in the question would be 1.0---no calculation required. However, if it had said "During 50 draws, exactly 40 blue and 10 white were observed", the question would be more meaningful and would require some computations.

 

[BvU]

That seems to say that you look into the urn after all the draws have been made, and you see 40 blue and 10 white; that is what the word "after" says to me

advocate of the dark side. My (naive?) perception is that the 40 / 10 is the outcome (i.e. that what came out of the urn) and you were not allowed to look if the observed marbles were put back or not.

I'm still waiting for @Tajeshwar to come up with the probability of this 40 / 10 outcome for the case that the marbles were put back after observing the colour. Because I do not agree with

The .5 prob of picking either with replacement or without

In the original story there was no coin toss before the drawing of marbles started.

 

[SammyS]

Yes thank you. This was a big part of what was confusing me. The .5 prob of picking either with replacement or without.

On the first draw, what is the probability of picking:

a blue marble?
a white marble?
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