You have 11 marbles in a bag. 4 of them are black and 7 of them are white.

Randomly give 5 marbles to Daniel and the remaining 6 marbles to Jeff.

Let X be the number of black marbles Daniel gets, and Y the number of black marbles Jeff gets.

What is the joint probability mass function of (X,Y)? Please provide a function and/or table.

Alice rolls a pair of fair six-sided dice until a sum of 6 appears for the first time. Independently, Bob rolls two fair six-sided dice until he rolls a sum 7 for the first time. Find the probability that the number of times Alice rolls her dice is equal to or within one of the number of times Bob rolls his dice.

Q. Assume that you randomly choose an integer from 1, 2, 3 and 4, and next roll a fair four-sided
die until you get an outcome that is larger than or equal to the randomly chosen integer.
(a) What is the probability mass function of the number of times you will roll the die?
(b) What is the expected value of the number of times you will roll the die?

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with lambda the expected number of trees per acre given by 80.

(a)(3pts) What is the probability that in a certain quarter acre plot there will be at most 16 trees?

(b)(2pts) If the forest covers 85,000 acres what is the expected number of trees in the forest?

Please provide the answer in Rstudio code. Thank you.

A company has discovered that a recent batch of batteries had manufacturing​ flaws, and has issued a recall. In a group of 15 batteries covered by the​ recall, 3 are dead. Two batteries at random are chosen from the package of 15

​b) Create a probability model for the number of good batteries chosen.

​c) What's the expected number of good​ batteries?

​d) What's the standard​ deviation?

The probability of winning in a certain state lottery is said to be about 1/9. If it is exactly 1/9, what a random variable represents the distribution of the number of tickets a person must purchase up to and including the first winning ticket? Plot the PMF of this random variable. the distribution of the number of tickets purchased up to and including the second winning ticket can be described by what distribution?

of nine executives in a business firm, four are married, three have never married, and two are divorced. three of the executives are to be selected for promotion. Let Y1 donate the number of married executives and Y2 donate the number of never married executives among the three selected for promotion. Assuming that three are randomly selected from nine available find the joint probability function of Y1 and Y2.

1) We are creating a new card game with a new deck. Unlike the normal deck that has 13 ranks (Ace through King) and 4 Suits (hearts, diamonds, spades, and clubs), our deck will be made up of the following.

Each card will have:
i) One rank from 1 to 16.
ii) One of 5 different suits.

Hence, there are 80 cards in the deck with 16 ranks for each of the 5 different suits, and none of the cards will be face cards! So, a card rank 11 would just have an 11 on it. Hence, there is no discussion of "royal" anything since there won't be any cards that are "royalty" like King or Queen, and no face cards!

The game is played by dealing each player 5 cards from the deck. Our goal is to determine which hands would beat other hands using probability. Obviously the hands that are harder to get (i.e. are more rare) should beat hands that are easier to get. a) How many different ways are there to get any 5 card hand?
The number of ways of getting any 5 card hand is
DO NOT USE ANY COMMAS

b)How many different ways are there to get exactly 1 pair (i.e. 2 cards with the same rank)?
The number of ways of getting exactly 1 pair is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 1 pair?
Round your answer to 7 decimal places.


c) How many different ways are there to get exactly 2 pair (i.e. 2 different sets of 2 cards with the same rank)?
The number of ways of getting exactly 2 pair is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 2 pair?
Round your answer to 7 decimal places.


d) How many different ways are there to get exactly 3 of a kind (i.e. 3 cards with the same rank)?
The number of ways of getting exactly 3 of a kind is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 3 of a kind?
Round your answer to 7 decimal places.


e) How many different ways are there to get exactly 4 of a kind (i.e. 4 cards with the same rank)?
The number of ways of getting exactly 4 of a kind is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 4 of a kind?
Round your answer to 7 decimal places.


f) How many different ways are there to get exactly 5 of a kind (i.e. 5 cards with the same rank)?
The number of ways of getting exactly 5 of a kind is
DO NOT USE ANY COMMAS

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 231 customers on the number of hours cars are parked and the amount they are charged.

a-1. Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.)

a-2. Is this a discrete or a continuous probability distribution?

b-1. Find the mean and the standard deviation of the number of hours parked. (Do not round the intermediate calculations. Round your final answers to 3 decimal places.)

b-2. How long is a typical customer parked? (Do not round the intermediate calculations. Round your final answer to 3 decimal places.)

c. Find the mean and the standard deviation of the amount charged. (Do not round the intermediate calculations. Round your final answers to 3 decimal places.)

An exhaustive study of all active Facebook accounts was recently conducted by Facebook. One variable Facebook recorded was the number of friends X of each Facebook user. Suppose X has expected value E(X) = 187 and standard deviation SD(X) = 283.6. Since the possible values of X are only integers and since the distribution of X is highly skewed to the right, the distribution of X cannot be described by a normal model. Suppose you select a random sample of 35 Facebook users and record the number of Facebook friends each user has.

Question 1. What is the probability that the 35 Facebook users in your sample have a sample mean number of friends x greater than 168?

(use 4 decimal places in your answer)

Question 2. What is the probability that the 35 Facebook users in your sample have a sample mean number of friends x less than 197?

(use 4 decimal places in your answer)

Question 3. The Central Limit theorem was needed to answer questions 1 and 2 above.

True or False?