What are Some Questions on Probability Distributions?

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Discussion Overview

The discussion revolves around several probability distribution questions, including scenarios involving a carnival game with a penny, random chords in a circle, rolling dice, and matching cards from two decks. Participants seek clarification and solutions to these problems, which span theoretical and applied aspects of probability.

Discussion Character

  • Homework-related
  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant describes the conditions for winning in the carnival game, suggesting the probability of winning is 1/16 based on the penny's placement within a square.
  • Another participant raises concerns about ambiguity in the second question regarding the random selection of chords and proposes different interpretations that could lead to varying results.
  • For the third question, a participant notes that determining the probability distribution for rolling a die three times is complex and suggests it should be straightforward, though they do not provide a detailed solution.
  • A hint is provided for the fourth question, indicating the number of configurations for matching cards in the second deck, but does not resolve the question.
  • One participant expresses frustration and confusion over the questions, indicating a desire for clearer explanations.
  • Another participant asserts that the probability for the first question is zero, characterizing it as a carnival game, but does not clarify the reasoning behind this claim.
  • A participant encourages the original poster to clarify their confusion regarding a previous response, particularly about the second question, and references Bertrand's Paradox as a related topic.

Areas of Agreement / Disagreement

There is no consensus on the solutions to the posed questions. Participants express differing views on the interpretations and calculations involved, particularly for the second question, and the discussion remains unresolved.

Contextual Notes

Participants note ambiguities and assumptions in the questions, particularly regarding the definitions of randomness and the conditions for the chord selection in the second question. The complexity of the probability distributions for the die rolls and card matches is also acknowledged.

kikar
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hey guys..

Im having trouble answering the following questions I tried everything I can. I JUSS don't get it

1)In a common carnival game the player tosses a penny from a distance of about 5 feet onto a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

2) 2If a chord is selected at random on a fixed circle, what is the probability that its length exceeds the radius of the circle?

3a) You roll a single die 3 times. Let X be the random variable which measures how many of each roll turns up (e.g. 3 one’s, 2 two’s, 1 three, 1 four, 1 six). Determine the probability distribution of the situation.
b) The above random variable is said to exhibit a multinomial distribution. Use the above question to derive an expression (equation) for a particular outcome of X.

4) From a shuffled deck, cards are laid out on a table one at a time, face up from left to right, and then another deck is laid out so that each of its cards are beneath a card from the first deck. What is the probability that exactly ‘r’ matches occur (a match being the same value and suit matching)

please help me out, I am stuck, i got the other 16 questions right, and i am not satisfied with just that, because I am determined to find out what i did wrong.
thanks
 
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1. The center of the penny (in a square) has to be 3/4 in. or more away from any edge. Therefore the center has to be inside a 1/4 in. by 1/4 in. square in the middle of the 1 in. square. Assuming uniform, the answer is 1/16.

2. There may be an ambiguity. When you say random, what is random and what is its distribution?
A possible distribution is the center of the chord is uniform within the circle and the chord is perpendicular to the radius at the center.
Another would be one end of the chord is fixed and the other end is uniform on the edge of the circle.
I haven't worked it out, but I suspect the results would be different.

3. and 4. Too much work. Should be straightforward.
 
Hin for 4: There are 52! possible configurations of the second card deck.
If we select r cards from the above deck, there are (52-r)! configurations of the lower deck which have those r cards matching. (do you see why?)
 
oh man
Im confused even more!

these questions are driving me insane, i want to know how to do them :(
 
ne oen? :-p
 
1) is definitely zero, since it's a carnie game.

Not sure about the others.
 
Kikar,
read mathmans post carefully again and post exactly what u did not understand in his post.

As regards to question 2, mathman is correct ...
Its an old problem which has become a classic on its own and is known as the Bertrands Paradox. To read more on this go here,
http://mathworld.wolfram.com/BertrandsProblem.html

-- AI
 

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