# Probability question - balls in urn (hypergeometric?)

• dizzle1518
In summary, the problem is asking for the probability that the last ball drawn from an urn containing 5 black and 8 red balls is black, given that the first ball drawn was red. This can be computed in multiple ways, including using the hypergeometric distribution or by considering the probabilities of each ball being black in a simpler manner. Ultimately, the probability is 5/12.
dizzle1518
Hi all,

I need help with the following problem:

The urn contains 5 black and 8 red balls. You close your eyes and
remove balls from the urn one by one without replacement. What is
the probability that the last ball is black?

This looks to me like it is a hypergeometric distribution problem. I set it up in the following way: ((5 choose 4)*(8 choose 8))/(13 choose 12). In order for the last ball to be black we have to remove 12 balls such that the remaining 13th one is black. so we have 5 choose 4 orderings of the black balls and 8 choose 8 orderings of the red balls. Since we are taking out a total of 12 balls then there are 13 choose 12 possible orderings or the red and black balls.

Is this correct?

Thanks,
--David

It's correct, but it's the complicated way to compute the answer. What do you get when you do it that way? Now think about it a different way. What's the probability the first ball is black? What's the probability the second ball is black? What's the probability that ANY ball is black?

Dick said:
It's correct, but it's the complicated way to compute the answer. What do you get when you do it that way? Now think about it a different way. What's the probability the first ball is black? What's the probability the second ball is black? What's the probability that ANY ball is black?

would the answer to a modified version of this problem be 5/12?

The urn contains 5 black and 8 red balls. You close
your eyes and remove balls from the urn one by one without replacement.
What is the probability that the last ball is black given that the 1st ball
is red?

dizzle1518 said:
would the answer to a modified version of this problem be 5/12?

The urn contains 5 black and 8 red balls. You close
your eyes and remove balls from the urn one by one without replacement.
What is the probability that the last ball is black given that the 1st ball
is red?

Sure it is.

## What is the Hypergeometric distribution?

The Hypergeometric distribution is a probability distribution that describes the number of successes in a sequence of draws without replacement from a finite population. It is commonly used in situations where the population size is small and the probability of success changes from trial to trial.

## How is the Hypergeometric distribution related to the balls in urn problem?

The Hypergeometric distribution is often used to model the probability of selecting a certain number of colored balls from an urn without replacement. This is because the number of successes (colored balls) in a sample (draws from the urn) follows a hypergeometric distribution.

## What are the key components of a Hypergeometric distribution?

The key components of a Hypergeometric distribution are the population size (N), the number of successes in the population (K), and the sample size (n). These values are used to calculate the probability of obtaining a specific number of successes in a sample without replacement.

## How is the Hypergeometric distribution different from the Binomial distribution?

The Hypergeometric distribution and the Binomial distribution are both used to model the probability of obtaining a certain number of successes in a sample. The main difference is that the Hypergeometric distribution is used when the sample size is small and the probability of success changes from trial to trial, while the Binomial distribution is used when the sample size is large and the probability of success remains constant.

## Can the Hypergeometric distribution be used for calculating probabilities in other situations?

Yes, the Hypergeometric distribution can be used in other situations where there is a finite population and the probability of success changes from trial to trial. For example, it can be used to model the probability of selecting a certain number of defective items from a production line or the probability of winning a certain number of games in a series.

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