Probability Mass Function

An urn contains 10 white chips and b blue chips. A chip is drawn at random and then is returned to the urn along with 2 chips of the same color. Let $\displaystyle X_n$ be the number of white chips drawn during the first $\displaystyle n$ draws. Find the probability mass function of $\displaystyle X_2$ and $\displaystyle E([X_2])$. Show all your work.

Re: Probability Mass Function

Hey videomagru.

The first thing you should do is figure out the probability function of the next draw given the results of the first.

This problem is known as a Markov problem since the next probability distribution depends only on what happened in the last draw.

Are you familiar with Markov Chains?

Re: Probability Mass Function

Quote:

Originally Posted by

**chiro** Hey videomagru.

The first thing you should do is figure out the probability function of the next draw given the results of the first.

This problem is known as a Markov problem since the next probability distribution depends only on what happened in the last draw.

Are you familiar with Markov Chains?

No I have not heard of Markov Chains. Do they apply here?

Re: Probability Mass Function

Quote:

Originally Posted by

**chiro** Hey videomagru.

The first thing you should do is figure out the probability function of the next draw given the results of the first.

This problem is known as a Markov problem since the next probability distribution depends only on what happened in the last draw.

Are you familiar with Markov Chains?

We are not currently studying anything called Markov Chains or the Markov Problem. I did just do a very similar problem that followed Polya's Urn Model, i.e., An urn contains w white chips and 10 blue chips. A chip is drawn at random and then is returned to the urn along with 2 chips of the same color. Let $\displaystyle W_n$ be the event that the $\displaystyle n^{th}$ draw is white and $\displaystyle B_n$ be the event that the $\displaystyle n^{th}$ draw is black. Show all of your work in computing: P($\displaystyle W_3$) and P($\displaystyle B_3$). But I don't think that can be applied here?

Re: Probability Mass Function

The Markov chain solution is very easy to setup and you can get all the probabilities once you have setup the transition matrix.

If you haven't covered it, then I should advise against it and see if you can use the Urn problem format to construct the distribution.

The power with the Markov chain approach is that it is used for any situation much like you have where probabilities only change based on what just happened and there are many problems like this in both theory and practice which is why it is so useful to consider them.