The inside diameter of a piston ring is normally distributed with a mean of 10 cm and a standard deviation of 0.04 cm.

1. What is the probability that a piston ring will have an inside diameter less than 10.075 cm?
2. What is the probability that a piston ring will have an inside diameter between 9.94 and 10.045 cm?
3. What proportion of rings will have inside diameters exceeding 10.066 cm?
4. Suppose that five piston rings are randomly selected. What is the probability that the diameters of exactly four of the five will exceed 10.066 cm?
5. You continue selecting piston rings from a large number until you find 3 that exceed 10.066 cm. What is the probability that you check exactly 15 rings?

23. NOTE: Answers using z-scores rounded to 3 (or more) decimal places will work for this problem.

The population of weights for men attending a local health club is normally distributed with a mean of 175-lbs and a standard deviation of 26-lbs. An elevator in the health club is limited to 32 occupants, but it will be overloaded if the total weight is in excess of 5952-lbs.

a. Assume that there are 32 men in the elevator. What is the average weight beyond which the elevator would be considered overloaded?
average weight = lbs

b. What is the probability that one randomly selected male health club member will exceed this weight?
P(one man exceeds) =  
(Report answer accurate to 4 decimal places.)

c. If we assume that 32 male occupants in the elevator are the result of a random selection, find the probability that the elevator will be overloaded?
P(elevator overloaded) =  
(Report answer accurate to 4 decimal places.)

d. If the elevator is full (on average) 6 times a day, how many times will the elevator be overloaded in one (non-leap) year?
number of times overloaded =  
(Report answer rounded to the nearest whole number.)

The population of weights for men attending a local health club is normally distributed with a mean of 169-lbs and a standard deviation of 30-lbs. An elevator in the health club is limited to 34 occupants, but it will be overloaded if the total weight is in excess of 6290-lbs.

Assume that there are 34 men in the elevator. What is the average weight of the 34 men beyond which the elevator would be considered overloaded?
average weight =  lbs

What is the probability that one randomly selected male health club member will exceed this weight?
P(one man exceeds) =
(Report answer accurate to 4 decimal places.)

If we assume that 34 male occupants in the elevator are the result of a random selection, find the probability that the elevator will be overloaded?
P(elevator overloaded) =
(Report answer accurate to 4 decimal places.)

If the elevator is full (on average) 7 times a day, how many times will the elevator be overloaded in one (non-leap) year?
number of times overloaded =

harmonic if the period (the time for one cycle) does not depend on the amplitude (the maximum displacement from equilibrium). In the following set, identify the oscillations that are simple harmonic, the ones that are approximately simple harmonic, and the ones that are not simple harmonic. Please comment on why/how you make your identifications.

1. The pendulum in a grandfather clock

2. A boat in water pushed down and released

3. A child on a swing

4. A mass hanging from a spring

5. A ping pong ball bouncing on the floor.

Two snooker players, Player A and Player B, play a match of a ‘best-of-seven’ frames of snooker, i.e. the first player to win 4 frames wins the match. The probability Player A wins a frame is 0.55 and the probability Player B wins a frame is therefore 0.45.

(a) What is the probability that Player A wins the match by a score of 4-2?
(b) What is the overall probability Player B wins the match?
(c) What is the expected number of frames played in a match?

Hello there, I'm wondering can you do this problem without arrays? Using the given code on C++ Platform.

Let me know ASAP.

#include <iostream>

#include <time.h>

using namespace std;

void shoot(bool &targetAlive, double accuracy)

{

double random = (rand() % 1000) / 1000.;

targetAlive = !(random < accuracy);

}

int startDuel(int initialTurn)

{

bool personAlive[3] = {true, true, true};

double accuracy[3] = {0.333, 0.5, 1.0};

int turn = initialTurn; // which person has to shoot, initialTurn represents the first person who shoots

int aliveCount = 3; // total persons still alive

while (aliveCount > 1) { // loop until only one person is alive

if(!personAlive[turn]) { // if person is dead

turn = (turn + 1) % 3; // give turn to next person

continue;

}

int highestAccuracyPersonAlive = -1;

int highestAccuracy = -1;

for (int i = 0; i < 3; i++)

{

if (i != turn && personAlive[i] && accuracy[i] > highestAccuracy) { // person has the highest accuracy and is alive, so far

highestAccuracyPersonAlive = i;

highestAccuracy = accuracy[i];

}

}

// shoot the person with the highest accuracy and who is still alive

shoot(personAlive[highestAccuracyPersonAlive], accuracy[turn]);

if(!personAlive[highestAccuracyPersonAlive])

aliveCount--; // decrease alive count if person shot is dead

turn = (turn + 1) % 3; // give the turn to shoot to the next person

}

if(personAlive[0])

return 0;

if(personAlive[1])

return 1;

return 2;

}

int main() {

srand((unsigned) time(NULL));

int wins[3] = {0, 0, 0};

for (int i = 0; i < 1000; i++)

{

wins[startDuel(0)]++;

}

cout << "Probability of Aaron winning: " << wins[0]/1000. << endl;

cout << "Probability of Bob winning: " << wins[1]/1000. << endl;

cout << "Probability of Charlie winning: " << wins[2]/1000. << endl;

wins[0] = wins[1] = wins[2] = 0;

for (int i = 0; i < 1000; i++)

{

// Counterintuitive strategy is equivalent to giving the first turn to shoot to Bob (person with index 1)

wins[startDuel(1)]++;

}

cout << "Probability of Aaron winning with counterintuitive strategy: " << wins[0]/1000. << endl;

cout << "Probability of Bob winning counterintuitive strategy: " << wins[1]/1000. << endl;

cout << "Probability of Charlie winning counterintuitive strategy: " << wins[2]/1000. << endl;

return 0;

}

Consider a second price auction for a single item with two bidders. Suppose the bidders have independent private values, uniformly drawn in the interval [0, 1]. Suppose the seller sets a reserve price p = 0.5; that is, only bids above p = 0.5 can win. If a bidder bids above p and the other bids below p, then the first bidder wins and pays a price p. If both bid above p, then the highest bidder wins and pays the second highest price.

In the Bayesian equilibrium in undominated strategies, what is the probability that the item will not be sold?

:)

Alice and Bob play the following game: in each round, Alice first rolls a single standard fair die. Bob then rolls a single standard fair die. If the difference between Bob’s roll and Alice's roll is at most one, Bob wins the round. Otherwise, Alice wins the round.

(a) (5 points) What is the probability that Bob wins a single round?

(b) (7 points) Alice and Bob play until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series?

(c) (7 points) In a single series, what is the expected number of wins for Bob?

(d) (6 points) In a single series, how many more games is Alice expected to win than Bob? That is, what is the expected value of the number of wins for Alice minus the number of wins for Bob?

(e) (5 points) In a single series, what is the variance of the expected number of wins for Bob?

PLEASE DO NOT COPY OTHERS ANSWER, THANK YOU!

Alice and Bob play the following game: in each round, Alice first rolls a single standard fair die. Bob then rolls a single standard fair die. If the difference between Bob’s roll and Alice's roll is at most one, Bob wins the round. Otherwise, Alice wins the round.

(a) (5 points) What is the probability that Bob wins a single round?

(b) (7 points) Alice and Bob play until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series?

(c) (7 points) In a single series, what is the expected number of wins for Bob?

(d) (6 points) In a single series, how many more games is Alice expected to win than Bob? That is, what is the expected value of the number of wins for Alice minus the number of wins for Bob?

(e) (5 points) In a single series, what is the variance of the expected number of wins for Bob?