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Old 04-09-2013, 02:15 PM
silvrous silvrous is offline
Join Date: Apr 2012
Posts: 24
Default Re: *ANSWER* Homework1 - Q3

Originally Posted by jcmorales1564 View Post
Hello. This is my first post in the forum. I am trying very hard to maintain the pace in this course. I lost interest at first since I thought that I needed to have the book to participate in the forum but I received an email from the instructor that indicated the answer to the login question (thanks!). I finished the Data Analysis course in Coursera (Dr. Jeff Leek) a couple of weeks ago (that is where I became interested in Machine Learning) and the forum was one of the highlights. I am glad that I have finally been able to join this forum.

At any rate, in this problem I initially thought that p = 1/2 but this answer kept bugging me... I kept recalling the famous Monty Hall three-door scenario where p = 2/3. Then I recalled the concept of sample space, i.e., the set of all possible outcomes, and I was able to finally visualize the problem properly.

Define the bags and the balls as follows:

B = Black Ball
W = White Ball
BBbag = bag with 2 black balls
BWbag = bag with one black and one white ball

There are three actions that take place:
1. Select a bag
2. Select the first ball from the bag
3. Select the second ball from the bag

Separating each action by a hyphen, then the set of possible outcomes or Sample Space is:
  1. BWbag - W - B
  2. BWbag - B - W
  3. BBbag - B - B
  4. BBbag - B - B

The condition set by the problem is that the first ball that is chosen must be black so only the last three cases in the sample space apply. Of these three cases, only two meet the outcome (second ball is black).

So, defining probability as the ratio of the number of equally likely outcomes that produce a given event to the total number of possible outcomes that meet our condition, then the probability that the second ball is black is 2/3.
Wow, that does make sense and it's way easier than my Bayesian solution!
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