1. Desta is a gambler and regularly plays several rounds of a gamble in which he wins $6,000 if odd number of dots face up when a fair die is rolled twice and loses $2,000 for any other outcome from the two rolls of the die. In other words, the outcome of the gamble is determined by rolling a die twice on each round of the gamble and Desta wins only if odd number of dots show up on top in both rolls of the die and loses if any other outcome occurs. Desta is considering playing 18 rounds of such a gamble next week hoping that he will win back the $1,000 he lost in a similar gamble last week. If we let X represent the number of wins for Desta out of the next 18 rounds of the gamble, X will have the binomial probability distribution.

a. Please calculate the probability of success for Desta on each round of the gamble. Show how you arrived at your answer. [Hint: See the joint probability rule for independent events on slide 14 of PPT_SLIDES_SECOND_WEEK).                                           (1 point)

b. What is the probability that Desta will win none of the 18 rounds of the gamble?   (1 point)

c. What is the probability that Desta will lose at least 10 of the 18 rounds of the gamble? Show your work.                                                            (1 point)

d. What is the probability that Desta will win fewer than 8 out of the 18 rounds of the gamble? Show your work.                                                    (1 point)

e. What is the probability that Desta will lose at most 12 out of the 18 rounds of the gamble? Show your work.                                                    (1 point)

f. What is the probability that Desta will lose more than half out of the 18 rounds of the gamble?    Show your work.                                                       (1 point)                                                                                       

  g. How much money is Desta expected to win in the 18 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Desta is playing a smart gamble? Please show how you arrived at your results and explain your final answer.                                                            (2 points)

h. Calculate and interpret the standard deviation for the number of wins for Desta in the next 18 rounds of the gamble. Show your work.                   (1 point)

"Einstein riddle" The riddle asked us to complete a table of answers in a way that ensured various constraints were satisfied. A "correct" answer to the riddle was a completed table that satisfied all the constraints provided. The table was 5 rows by 5 columns. For each row you could pick the order of 5 parameters - for example, the order of 5 different colours or 5 different car models. In this particular problem there were no duplicate assignments - i.e. you couldn't have two cars of the same colour or make.

Would enforcing less constraints have made our program find a solution faster or slower. Why ? . How large was the search space for this problem before we applied constraints ? (i.e. how many different ways were there to complete the table) . Explain why understanding the size of the search space is one of the first things you should do when investigating a new problem . How would allowing duplication on the colour of the cars affected the problem ?

In a championship series, the winner emerges by winning 5 out of eight games. Teams A and B play in the championship, but team A has a 60% chance of winning compared to team B.

a.Find the probability that team A wins the championship

b.What is the probability that team A wins in 7 games.

c.If there’s a playoff series in which the winner emerges by winning 2 out of 3 games, what is the probability that team A wins the series.

Assignment

Write a network diagnostics bash script that does the following:

• Finds the IPv4 address of the default gateway on your machine (one way to get this is the 'default' entry, also shown as 0.0.0.0, in the output of 'ip route') and runs a ping (with a count of 5 pings) to it.
• Runs another count of 5 pings to the site example.com.
• Outputs a 1-line summary of interfaces on your machine, not including the loopback address (lo).
• Outputs a 1-line summary of how many TCP ports are open (in state LISTEN)

The output of your script should look exactly like this (different numbers are OK). For the ping times output, use the 'avg' number from the ping command. This is the average that is calculated for you -- no need to calculate it in your script.

Ping time to default gateway:  0.348 ms
Ping time to example.com: 24.807 ms
Interface count:  2, and 2/2 are up
There are 7 TCP ports in the LISTEN state

Please solve the riddle, how can a virus encode a protein larger than the size of its genome?

1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players take turns. When it is a player's turn, she can remove any number of matches from either pile. Each player is required to remove some number of matches if either pile has matches remaining, and can only remove matches from one pile at a time. Whichever player removes the last match wins the game. Winning gives a player a payoff equal to 1, and losing gives a player a payoff equal to 0. The initial configuration of the piles has one match in one of the piles, and two in the other one. a) Write down the game tree for this sequential game. [3 points] b) What is the Nash equilibrium? [3 points]

An elevator has a placard stating that the maximum capacity is 1780—10 passengers.​ So, 10 adult male passengers can have a mean weight of up to 1780 divided by 10 equals 178 pounds. 1780/10=178 pounds. If the elevator is loaded with 10 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 178lb.​ (Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of29 lb.)

Does this elevator appear to be​ safe?

A game works as follows: each player rolls one dice (let's call it ''n''). After both players have made the first roll, they have the choice to leave or not. If a player exits, he automatically looses; if not, he rolls a second time (let's call this second roll "m'') and receives n^m points . The player with the most points wins, in case of a tie no one wins.
a) Give the fundamental space of the two throws.
b) Give the fundamental space of the number of points obtained.
c) Show that a player should automatically withdraw if n = 1.
e) Let n₁ and n₂ be the value of n of the first and second player (respectively). Knowing that n₁ = 2 and n₂ = 3, what is the probability that the first player wins?
d) Knowing that the first player will pull out automatically if n₁ = 1, half the time if n₁ < n₂ and never otherwise, what is the probability that the first player will win?
probability of the first player pulling out?

There is 2 players in the game.

A group of n people get on an elevator at Floor 0. There are m floors besides Floor 0. Each person uniformly randomly chooses one of those m floors to stop at. (All choices are independent, and no one gets on the elevator after Floor 0.) The elevator stops at a floor if at least one person has chosen to stop there. Let X be the number of stops that the elevator makes after Floor 0.

(a) What is E[X]? Var(X)?

A deck of 10 cards consists of 3 hearts cards (♥) cards, 3 diamonds (♦) cards, and 4 spades (♠) cards. The deck is shuffled until the cards are in random order, and then it is determined (without any reshuffling) whether the following three gamblers have won their bets. Alice bets that the 3 hearts cards are together (next to each other, in any order) in the deck, Bob bets that the 3 diamonds cards are together in the deck, and Carol bets that the 4 spades cards are together in the deck. If Alice wins her bet, she gets $5, if Bob wins his bet he also gets $5, while Carol gets $20 if she wins her bet. Nobody wins or loses money on a lost bet.

(a) Compute the probability that Alice wins her bet, the probability that Bob wins his bet, and the probability that Carol wins her bet. Give the three answers as reduced fractions.

(b) Compute the probability that all three gamblers win their bets.

(c) Compute the expected combined dollar amount that the three gamblers get in this game. Give the answer as a reduced fraction.