Probability Question - Urn problem

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In summary, the problem presents an urn with 4 white and 4 black balls. The objective is to randomly select 4 balls until exactly 2 of them are white. The probability of making exactly n selections can be calculated using the formula (4C2)(4/8)^2(1-4/8)^2, where 4C2 represents 4 choose 2. However, this formula assumes that the balls are picked simultaneously, which is not the case. The probability of choosing a white ball changes after each selection, making the initial formula incorrect. In order to accurately calculate the probability, we must consider the sample space and the possibility of succeeding on the first, second, or subsequent trials.
  • #1
GreenPrint
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Homework Statement



An urn contains 4 white and 4 black balls. We randomly choose 4 balls. IF 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. this continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly n selections.

Homework Equations





The Attempt at a Solution



I don't see why I can't find the probability of drawing exactly 2 white balls using this method.

(4C2)(4/8)^2(1-4/8)^2
where 4C2 is 4 choose 2

apparently this is wrong but I don't see why. Thanks for any help that you can provide.
 
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  • #2
GreenPrint said:

Homework Statement



An urn contains 4 white and 4 black balls. We randomly choose 4 balls. IF 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. this continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly n selections.

Homework Equations





The Attempt at a Solution



I don't see why I can't find the probability of drawing exactly 2 white balls using this method.

(4C2)(4/8)^2(1-4/8)^2
where 4C2 is 4 choose 2

apparently this is wrong but I don't see why. Thanks for any help that you can provide.

Well, why do you think it's right? The probability of choosing a white ball is only 1/2 the first time you choose a ball. It's not always 1/2, is it?
 
  • #3
oh i guess i always assumed you pick the things simultaneously as that's what I've been doing throughout the course so i guess this assumption is wrong?
 
  • #4
GreenPrint said:
oh i guess i always assumed you pick the things simultaneously as that's what I've been doing throughout the course so i guess this assumption is wrong?

If you pick a white ball first that means there are 3 whites left and 4 blacks. The odds of picking a second white or a black aren't 1/2 anymore. So yes, it's wrong.
 
  • #5
I think it is always useful to describe the sample space: you can succeed the 1st, or the 2nd time, etc.

EDIT: I mean, you can select the two balls in the first trial, or, if you don't, then at the second trial, etc.
 
Last edited:

Related to Probability Question - Urn problem

1. How does the urn problem relate to probability?

The urn problem is a classic example used in probability theory to illustrate the concept of probability. It involves drawing a sample from a set of objects in an urn, and calculating the probability of obtaining a certain outcome.

2. What is the basic setup of the urn problem?

In the urn problem, an urn contains a certain number of objects, some of which may be of different colors or have different properties. The goal is to calculate the probability of drawing a specific color or type of object from the urn.

3. What is the difference between sampling with replacement and sampling without replacement in the urn problem?

Sampling with replacement means that after each draw, the object is put back into the urn, and the total number of objects remains the same. Sampling without replacement means that the drawn object is not put back, and the total number of objects decreases with each draw.

4. Can the urn problem be applied to real-life situations?

Yes, the urn problem can be applied to many real-life situations, such as predicting the outcome of elections, estimating the success rate of a medical treatment, or calculating the chances of winning a lottery.

5. Are there any variations of the urn problem?

Yes, there are several variations of the urn problem, such as the multi-colored urn problem, the multiple urn problem, and the hidden urn problem. These variations may involve more complex scenarios, but the basic principles of probability still apply.

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