# Probability : Sampling without Replacement and 'Three Doors' Problem

1. Urn I and Urn II each contains 3 red and 3 white balls. First we transfer one ball from Urn I to Urn II. Then we transfer one ball from Urn II to Urn I. Finally we sample one ball from Urn and it is red. What is the probability the both transferred balls were also red?

2. In the famous "Monty Hall game" there are 3 doors. Monty Hall offers you the opportunity to win what is behind one of the three doors. Typically there was a really nice prize (ie. a car) behind one of the doors and a not-so-nice prize (ie. a goat) behind the other two. After selecting a door, Monty would then proceed to open one of the doors you didn't select. It is important to note here that Monty would NOT open the door that concealed the car. If he had a choice he would open the door on the left with probability p and the door on the right with probability 1 ? p. At this point, he would then ask you if you wanted to switch to the other door before revealing what you had won. Is it to your advantage to switch? (Hint: The answer might depend on p.)

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#### Solution Summary

Sampling without Replacement and the 'Three Doors' Problem are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.