Binomial Distribution

Tom and Jerry play one game per day. It is known that Tom’s winning chance is P[W] = 0.6 and if he does not win, then he loses. Game results are assumed to be independent. After FOUR days, variable T indicates a total number of games won by Tom. So J = 4 − T is the number of games he lost (or Jerry won).

Use the formula for binomial probabilities and probability rules to answer questions below.

1. Find the chance that Tom loses exactly one game,

   P (J = 1)

2. Evaluate probability that Tom wins at least two games,

   P [T ≥ 2]

3. Determine the chance than Jerry wins two or more games,

   P [J ≥ 2]

4. Find expected number of games won by Tom,

   E[T]

5. Use the formula for variance to determine the variance of T, that is

   Var[T]

6. Assume that after a game is over, the winner gets $1 from his partner. Thus the variable of interest is the balance for Tom, which is B = T − J. Evaluate the expected balance,
   E[B]

7. Derive the variance of B, that is Var[B]

Two team (A and B) play a series of baseball games. The team who wins three games of five-game-series wins the series. Consider A has home-field advantage (0.7 means A has probability of winning 0.7 if it plays in its field) and opponent-field disadvantage (0.2 means A has probability of winning 0.2 if it plays in opponents field). If the series start on A team’s field and played alternately between A and B team’s fields, find the probability that series will be completed in four games.

Define the sample space. To get simpler expressions, you can prefer the set notations.
(a) (6 Points) What is the sample space of choosing a real number from interval [0, 1] which is larger than 0.5. Specify the properties of the sample space. Design two different random variables for this task.
(b) (6 Points) You play rock-paper-scissors with your friend. The one with total three wins or two subsequent wins will be the winner of the game. Give a probability tree of you being the winner. State the sample space by using probability tree diagram.
(You can use nested probability trees to make the representation easier. For example, you can generate a tree only for win-loss possibilities (without any information about rock-paper-scissors), one another tree for Win, including all possible cases with rock- paper-scissors, similarly one another tree for Loss, again including all possible cases with rock-paper-scissors)

1. Flip a fair coin ten times. Find the probability of at least seven heads.

2. Draw five cards at once from a deck. Find the probability of getting two pairs.

3. Roll a die infinitely times. Find the probability that you see an even number before you see an one.

4. You and your friend take turns to draw from an urn containing one green marble and one hundred blue marbles, one at a time and you keep the marble. Whoever draw the green marble wins. Suppose you draw first. What is the probability that you win?

Question 1:

Jaycar, an electronics store sponsors the Canterbury Bulldogs, a team that competes in the NRL. An individual store owner is looking to promote the sale of their 160W solar panels and is planning to offer a discount on the price of the panels from Monday to Friday based on the number of home games won in the past weekend of NRL. The owner of the store has asked us to do some analysis of the promotion assuming the following:

(a) The number of wins each weekend out of eight games is a binomial random variable with a probability of the home team winning being equal to p.

(b) The number of solar panels normally sold from Monday to Friday is a Poisson random variable with mean units sold equal to λ.

(c) The number of wins in a weekend for home teams in the NRL and the number of solar panels sold from Monday to Friday are independent.

Question 1 A):

If we let X represent the number of home team wins out of eight games in a weekend of NRL, what will the mean and variance of X be in terms of p.

Question 1 B):

If we let Y represent the number of 160W solar panels sold from Monday to Fridays, what will the mean and variance of Y be in terms of λ.

Question 1C):

The “cost” of the promotion will be a new random variable that is a function of:

the number of home wins over the weekend;

the number of solar panels that will be sold; and the discount offered per game. If we let the cost be a new random variable such that Z = a*X*Y , find the mean and variance of Z. Hint: The variables are assumed to be independent.

The variance of the product of independent variables is given by

Var( XY ) = (E( X))^2*Var( Y ) + (E( Y ))^2*Var(X) + Var(X)*Var( Y ),

and Var(c*X) = c^2*Var(X).

What does the ​ ping command do? Please type "ping -c 4 8.8.8.8" and explain the
output. What is this "8.8.8.8" IP address? Please briefly explain what happens behind the
curtain when the ping command runs.

In the World Series the first team to win four games wins the championship. If the two teams are evenly matched, then the probabilities that the series will be decided after 4, 5, 6, or 7 games are given below. a) What is the expected number of games that will be required to decide the championship? b) Calculate the probabilities. Explain your logic and show all work.

Number of games: 4, 5, 6, 7

Probability: 1/8, 1/4, 5/16, 5/16

An individual buys 10 raffle tickets in hopes of winning one of 15 prizes to be given away by drawing tickets without replacement. The total number of raffle tickets sold is 168. Lt X be the number of prizes won by the individual.

A) Find the probability the individual wins at least one prize

B) Calculate the expected value E(X) accurate to 4 decimal places

C) Calculate the standard deviation SD(X) accurate to 4 decimal places

STAT15_2:

Arad has 9 different pairs of socks in the drawer - 6 of which are dark and 3 are bright. He removes two pairs of random socks from the drawer. Define the following random variables:
X - The number of pairs of dark socks that Ard issued.
Y - Receives the value 1 if the two pairs removed are dark or both light and the value 2 if one pair is dark and the other bright.

A. Find the joint probability function of X and Y and the marginal probability functions.
B. Are X and Y uncoordinated? Are they independent?
C. At least one pair of socks removed is known to be bright. What is the probability that a pair of dark socks was taken out?

In the game of craps, a pass line bet proceeds as follows: Two six-sided dice are rolled; the first roll of the dice in a craps round is called the “come out roll.” A come out roll of 7 or 11 automatically wins, and a come out roll of 2, 3, or 12 automatically loses. If 4, 5, 6, 8, 9, or 10 is rolled on the come out roll, that number becomes “the point.” The player keeps rolling the dice until either 7 or the point is rolled. If the point is rolled first, then the player wins the bet. If a 7 is rolled first, then the player loses. Write a program that simulates a game of craps using these rules without human input. Instead of asking for a wager, the program should calculate whether the player would win or lose. The program should simulate rolling the two dice and calculate the sum. Add a loop so that the program plays 10,000 games. Add counters that count how many times the player wins and how many times the player loses. At the end of the 10,000 games, compute the probability of winning [i.e., Wins / (Wins + Losses)] and output this value. Over the long run, who is going to win the most games, you or the house?