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.)

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.

Consider a standard deck of playing cards. You randomly select a card from the deck and find that you have drawn a face card. If you don't replace the card, what is the probability that the next card you draw will be a king?

A box of 9 gloves contains two left handed gloves and seven right handed gloves.
a: If two gloves are randomly selected from the box, withoutreplacement, what is the probability that both gloves will be right handed?
b: If two gloves are randomly selected from the box withoutreplacement, what is the probability that ther

Suppose that a batch of 100 items contains 6 that are defective and 94 that are nondefective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find
(a) P{X=0} and
(b) P{X>2}
(Answer first if sampling with replacing, and then if samplingwithout replacing)

For its validity, all hypothesis testing depends heavily on the assumption that the sample that is used was drawn using probabilitysampling techniques.
Why is this important?
What can you do if you just cannot use a probabilitysampling technique? (For example, suppose there is no good sampling frame available for the popul

The assets (in billions of dollars) of the four wealthiest people in a particular country are 47,35,28,11. Assume that the sample of size 2 are randomly selected with replacement from this population of four values.
a) After listing the possible samples and finding the mean of each sample, use the table to describe the sampli

Subject: Probability
Details: You appear on a game show and for your prize the host lets you choose one of three doors. Behind one door is a new car; behind each of the other doors is a goat. You choose a door. The host, who knows what's behind each door, then opens another door, which has a goat. He then asks if you want to p

What is the difference between probabilityand non-probabilitysampling? In the answer provide examples on the use of probabilitysamplingand non-probabilitysampling on real life problems.

A box contains 4 marbles: 2 red, one blue and one green. Two marbles will be selected at random. Find the probability of selecting each of the following.
A) with replacement B) withoutreplacement
i A red marble and then a blue marble
ii Two blue marbles
iii No red marbles
iv No green marbles.