What are Some Questions on Probability Distributions?

In summary, the conversation discusses various probability problems, including a carnival game with a penny, selecting a random chord on a circle, rolling a die multiple times, and shuffling cards from a deck. The conversation also expresses frustration and confusion with solving the problems. The summary also mentions a resource for further reading on one of the problems.
  • #1
kikar
8
0
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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  • #2
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.
 
  • #3
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?)
 
  • #4
oh man
Im confused even more!

these questions are driving me insane, i want to know how to do them :(
 
  • #5
ne oen? :-p
 
  • #6
1) is definately zero, since it's a carnie game.

Not sure about the others.
 
  • #7
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
 

FAQ: What are Some Questions on Probability Distributions?

What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes or values in a random experiment or event. It lists all possible outcomes and assigns a probability to each outcome, with the sum of all probabilities equal to 1.

What are the types of probability distributions?

The two main types of probability distributions are discrete and continuous. Discrete distributions are used for discrete random variables, which have a finite or countable number of possible values. Examples include the binomial distribution and the Poisson distribution. Continuous distributions are used for continuous random variables, which can take on any value within a specific range. Examples include the normal distribution and the exponential distribution.

What is the difference between a probability distribution and a probability density function?

A probability distribution is a function that assigns probabilities to specific outcomes or values, while a probability density function (PDF) is a function that represents the relative likelihood of different values occurring within a continuous range. In other words, a PDF is the continuous equivalent of a probability distribution for continuous random variables.

How is a probability distribution represented graphically?

A probability distribution can be represented graphically using a histogram or a line graph. In a histogram, the probability of each outcome is represented by a bar on a graph, while in a line graph, the probability is represented by a line connecting all possible outcomes.

What is the central limit theorem and how is it related to probability distributions?

The central limit theorem states that when a large number of independent and identically distributed random variables are added together, their sum will tend towards a normal distribution, regardless of the underlying distribution of the individual variables. This theorem is important in statistics as it allows us to use the normal distribution to approximate the behavior of many real-world phenomena.

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