If one egg is found to be good, then what is the probability that other is also good? A basket contains 10 eggs out of which 3 are rotten. Two eggs are taken out together at random. If one egg is found to be good, then what is the Probability that other is also good?



I applied conditional probability. It says that one of them is good, so the probability of the other one being good can be found in the 9 eggs left out of which 6 are good, so Probability = $6/9$

Am I right with my understanding?
Am I right with my understanding?


You're overlooking a subtle point here, one that is often overlooked. You do get the right result, but due to a bit of luck.

The problem is you're overlooking a choice &mdash; in addition to having selected a pair of eggs from the basket, you are also choosing to assign the labels "one" and "other" to the two eggs.

Correctly modeling how that choice is made extremely important.

Fortunately, the problem surely means for the choice to be made in the easier-to-understand fashion: either one of the following two equivalent models was used


The labels were assigned randomly, with each of the two choices having equal probability
The eggs were selected not as a pair, but one at a time, and the first egg selected being called "one" and the second egg selected being called "other"


The reason that this is surely right is that the problem appears to indicate either of the following typical procedures, which directly correspond to the two bullet points above:


Take a pair of eggs from the basket
Pick one of the two eggs
Check whether the chosen egg is good or rotten
Guess whether the other is good or rotten


or


Take one egg from the basket
Check whether it is good or rotten
Take another egg from the basket
Guess whether it is good or rotten




An example of a different procedure requiring a different model, incidentally, could be described as follows:


Take a pair of eggs from the basket
Your colleague checks whether they are good or rotten
If both are rotten, things stop here. Otherwise, your colleague tells you one is good
You ask your colleague to point to a good egg. 
You now guess whether the other egg is good or rotten.


(the actual problem being modeled doesn't have to include a colleague; that character can just be a fictional entity that enables the analysis to actually pick out a good egg)
I think it's $\frac{2}{7}$ if you pick them together at the same time and not put them back in the basket.