# Month: September 2017

# switching job offer envelopes puzzle (a.k.a. Monty Hall problem)

**Puzzle:
Three envelopes are presented in front of you by an interviewer. One contains a job offer, the other two contain rejection letters. You pick one of the envelopes. The interviewer then shows you the contents of one of the other envelopes, which is a rejection letter. The interviewer now gives you the opportunity to switch envelope choices. Should you switch?**

*Common solution:
The answer is yes. Say your original pick was envelope A. Originally, you had a 1/3 chance that envelope A contained the offer letter. There was a 2/3 chance that the offer letter was either in envelope B or C. If you stick with envelope A, you still have the same 1/3 chance. Now, the interviewer eliminated one of the envelopes (say, envelope B), which contained a rejection letter. So, by switching to envelope C, you now have a 2/3 chance of getting the offer and youâ€™ve doubled your chances.*

**Discussion: **

One critical piece of information is missing, and the answer actually depends on it. **Does the interviewer know the content of her envelopes, or opens one at random?**

**If the the interviewer knows the content of her envelopes, then the common solution is correct**. This is related to the so called**Standard assumptions (****wikipedia article on the Monty Hall problem****)****If the the interviewer does not know the content of her envelopes, then the common solution is not correct**. In this case we are dealing with conditional probabilities, given that the interviewer at random eliminated one of the rejection envelopes. This gives you a 50% conditional probability of success, and switching envelopes will not increase your chances.To make it more intuitive, consider the “law of large numbers” arguments. Suppose you play this random game 9 million times. Approximately 3 million times the interviewer at random will eliminate the offer letter. Out of the remaining approximately 6 million times, you will choose an offer letter about 3 million times. Your chance of success will not depend on switching envelopes (which is a great example on conditional probabilities).

This discussion is related to the **Monty Hall problem: The probability puzzle that makes your head melt (by the BBC)**

# Is the birthday paradox really a paradox ?

Links and References:

http://betterexplained.com/articles/u… (Regarding the birthday paradox)

http://www.bbc.com/news/magazine-2783… (more on the birthday paradox)

https://en.wikipedia.org/wiki/Birthda… (More on the birthday paradox)

http://dictionary.cambridge.org/dicti… (Paradox definition)

https://en.wikipedia.org/wiki/Pigeonh… (Pigeonhole principle)

# maximize the probability puzzle

You have 100 balls (50 black balls and 50 white balls) and 2 buckets. How do you divide the balls into the two buckets so as to maximize the probability of selecting a black ball, if 1 ball is chosen from 1 of the buckets at random?

# MathJax-LaTeX plugin to Aurora.

$$\begin{pmatrix}n\\k\end{pmatrix}=\frac{n!}{k!(n-k)!}=\frac{n\cdot(n-1)\cdot\ldots\cdot(n-k+1)}{k!}$$

\( E=mc^2 \)\( E=mc^2 \)

$$ i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right> $$

\[ i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right> \]

\( \LaTeX \)

\[ \LaTeX \]

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