# Probability of a given name be picked

10 names were put in a box. 2 names will be picked up randomly from it.

What's the probability of a given specific name be picked?

My opinion: 1/10 + 1/9. Other opinion: 1/10 + 1/10.

Small side-question: Does the probability is different when picking 2 names or picking one name and then another (with no reposition)?

If the probability equal in the above methods, what's the difference between those methods and the https://en.wikipedia.org/wiki/Monty_Hall_problem (Monty Hall problem).

Thank you!

$$\Pr\{\text{Choosing a specific name in 2 drawing out}\}=\\\Pr\{\text{Choosing the specific name in 1st drawing out}\}+\\\Pr\{\text{Choosing the specific name in 2nd drawing out}\}=\\{1\over 10}+{9\over 10}\cdot{1\over 9}=0.2$$

Given a specific name, there are $$9$$ pairs of names that contain it. The total number of pairs is $$45$$. Therefore the probability that a specific name will appear in a randomly chosen pair of names is $$\frac{1}{5}$$.

The slip of paper you chose has a $$\dfrac{1}{10}$$ chance of being correct while any of the other slips has a $$\dfrac{1}{9}$$ chance of being the correct name (because when you chose, you did not know about the incorrect slip, so there were 10 possible choices, but once Monty removed one, now there are only 9 possible choices--this includes the one you already picked--so any one you change to will have a greater probability of being correct).