Consider a joint PMF for the results of a study that compared the number of micro-strokes a patient suffered in a year (F) and an index (S) that characterizes the stress the person is exposed to. This PMF represents the probability of a randomly picked person from the studied population having F=f micro-strokes and S=s stress index.

a) The conditional PMF for the number of strokes F given stress index S=3.


b) The expected number of strokes and the variance of this magnitude for patients with S=3?


c) The conditional PMF for strokes and stress index given event A={(S,F) /s<3 and f<2}


d) There were 3000 patients in the study. How many you expect to find that have F and S in A (same A as above)?


e) What is the average stress index in this population? (hint: the marginal probability function above may be helpful)

?(a) If the grand prize is

?$12 comma 000 comma 00012,000,000?,

find and interpret the expected cash prize. If a ticket costs? $1, what is your expected profit from one? ticket?The expected cash prize is

?$0.30 0.30.

?(Round to the nearest cent as? needed.)

What is the correct interpretation of the expected cash? prize?

A.

On? average, you will win

?$0.300.30

per lottery ticket.

B.

You will win

?$0.300.30

on every lottery ticket.

C.

On? average, you will profit

?$0.300.30

per lottery ticket.The expected profit from one? $1 ticket is

?$negative 0.70 ?0.70.

?(b) To the nearest? million, how much should the grand prize be so that you can expect a? profit? Assume nobody else wins so that you do not have to share the grand prize.

?$nothing

?(c) Does the size of the grand prize affect your chance of? winning? Explain.

A.

?Yes, because your expected profit increases as the grand prize increases.

B.

?No, because the expected profit is always? $0 no matter what the grand prize is.

Ex. 2.40 European roulette.

The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their colour, they double their money. If it lands on another colour, they lose their money.

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?

(b) Suppose you bet $1 in three different rounds. What is the expected value and standard deviation of your total winnings?

(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

Ex. 2.34 Ace of clubs wins.

Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you win nothing. If you get a spade, you win $5. For any club, you win $10 plus an extra $20 for the ace of clubs.

(a) Create a probability model for the amount you win at this game. Also, find the expected winnings for a single game and the standard deviation of the winnings.

(b) What is the maximum amount you would be willing to pay to play this game? Explain your reasoning.

In a population of 10,000, there are 5000 nonsmokers, 2500 smokers of one pack or less per day, and 2500 smokers of more than one pack per day. During any month, there is a 5% probability that a nonsmoker will begin smoking a pack or less per day, and a 4% probability that a nonsmoker will begin smoking more than a pack per day. For smokers who smoke a pack or less per day, there is a 10% probability of quitting and a 10% probability of increasing to more than a pack per day. For smokers who smoke more than a pack per day, there is a 5% probability of quitting and a 10% probability of dropping to a pack or less per day. How many people will be in each group in 1 month, in 2 months, and in 1 year? (Round your answers to the nearest whole number.)

(a) in 1 month

(b) in 2 months

(c) in 1 year

Suppose the mean starting salary for nurses is $67,709 nationally. The standard deviation is approximately $10,970. Assume that the starting salary is normally distributed.

Find the probability that a starting nurse will make more than $88,000. Round to four decimals.
P(x > $88,000) =

Find the probability that a starting nurse will make less than $58,000. Round to four decimals.
P(x < $58,000) =

Find the probability that a starting nurse will make between $67,000 and $70,000. Round to four decimals.
P($67,000 < x < $70,000) =

What salary do 30% of all nurses make more than? Round to the nearest dollar.
$______

B) Suppose the mean yearly rainfall for a city is about 130 mm and the standard deviation is about 72 mm. Assume rainfall is normally distributed.

Find the probability that the yearly rainfall is less than 93 mm. Round to four decimal places.
P(x < 93) =

Find the probability that the yearly rainfall is more than 213 mm.
P(x > 213) =

Find the probability that the yearly rainfall is between 188 and 238 mm.
P(188 < x < 238) =

What rainfall amount are 90% of all yearly rainfalls more than? Round to the nearest whole number.

A recent study estimates that 45% of iPhone still have their phone their phone within 2 years of purchasing it. Suppose you randomly select 30 iPhone users. Let random variable X denote the number of iPhone users who still have their original phone after 2 years.

Describe the probability distribution of X (Hint: Give the name of the distribution and identify n and p).

Find the expected value of X. Round to 1 decimal place.

Find the standard deviation of X. Round to 3 decimal places.

Determine the probability that X equals 13. Round to 3 decimal places.

Determine the probability that EXACTLY 11 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

Determine the probability that at most 5 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

Determine the probability that at least 8 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

Determine the probability that between 6 and 9 iPhone users still have their original phone after 2 years. Round to 3 decimal places.

4/ A quality control inspector has determined that 0.25% of all parts manufactured by a particular machine are defective. If 50 parts are randomly selected, find the probability that there will be at most one defective part.

5/ A fair die is rolled 10 times. Compute the probability that a “one” appears exactly once.

6/ If two dice are tossed six times, find the probability of obtaining a sum of 7 two or three times.

7/ A machine produces parts of which 0.2% are defective. If a random sample of ten parts produced by this machine contains two or more defectives, the machine is shut down for repairs. Find the probability that the machine will be shut down for repairs based on this sampling plan.

8/ It was reported in a medical journal that about 70% of the individuals needing a kidney transplant find a suitable donor when they turn to registries of unrelated donors. Assume that a group of ten individuals needing a kidney transplant. Let x represent the number of individuals needing a kidney transplant who will find a suitable donor among the registries of unrelated donors.

a) Find the probability that all ten will find a suitable donor among the registries of unrelated donors.

b) Find the probability that at least eight will find a suitable donor among the registries of unrelated donors.

In a population of 10,000, there are 5000 nonsmokers, 2500 smokers of one pack or less per day, and 2500 smokers of more than one pack per day. During any month, there is a 6% probability that a nonsmoker will begin smoking a pack or less per day, and a 2% probability that a nonsmoker will begin smoking more than a pack per day. For smokers who smoke a pack or less per day, there is a 10% probability of quitting and a 10% probability of increasing to more than a pack per day. For smokers who smoke more than a pack per day, there is a 7% probability of quitting and a 10% probability of dropping to a pack or less per day. How many people will be in each group in 1 month, in 2 months, and in 1 year? (Round your answers to the nearest whole number.)

(a) in 1 month

(b) in 2 months

(c) in 1 year

You are rolling a fair dice 100 times.

• Let N be the number of times out of 100, when you roll an odd number. What is the most suitable distribution for N? Type one of the following (Poisson, binomial, normal,geometric)  .
• Determine the probability that the number of times N out of 100, when you roll an odd number is 60 or more P ( N ≥ 60 ) =  (use R and round to the third decimal place).
• Which distribution is the most suitabe to be used as an approximation? Type one of the following (Poisson, binomial, normal,geometric)  .
• Determine the expected value E ( N ) =  (integer) and the variance Var=  (integer) for the approximate distribution.

6. List the players that have either won more than 2 matches or lost more than 2 matches. List the player number, name, initials, won, and lost. Insert your snip here.

7. Which players are captains of a team and have ever served as a committee members? Display the player number, player name, initials, team number, and committee member position. This will require 3 tables. Insert your snip here.

8. Using a subquery, have any other players been penalized the same amount as player 8? Let MySQL do the work for you. Do not explicitly query for $25. Display the player number, player name, initials, and amount. Insert your snip here.

9. Who has played more than 2 matches? Display the player number as ‘Number’, concatenate the player name (initials, space, name) as ‘Name’, and the number of matches played as ‘Number of Matches’. Insert your snip here.

10. Sum the number of match wins and losses by player number. Display the player number as ‘Number’, concatenate the player name (initials, space, name) as ‘Name’, wins as ‘Total Wins’, and losses and ‘Total Losses’. Insert your snip here.

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Create database tennis;

USE tennis;

#--Create table players and fill it--------------------------

Create table players
(
playerno   int       not null    primary key,
name       varchar(15)   not null,
initials   varchar(3),
birth_date   date,
gender       char(1),
joined       int   not null,
street       varchar(15)   not null,
houseno       varchar(4),
zip       char(6),
town       varchar(10)   not null,
phoneno       char(10),
leagueno   char(4)
);

Insert into players values
(2,'Everett','R','1988-01-09','M',2000,'Stoney Road','43','3575NH','Stratford','070-237893','2411'),
(6,'Paramenter','R','1984-06-25','M',2002,'Haseltine Lane','80','1234KK','Stratford','070-476547','8467'),
(7,'Wise','GWS','1983-05-11','M',2006,'Edgecombe Way','39','9758VB','Stratford','070-347689',Null),
(8,'Newcastle','B','1982-07-08','F',2005,'Station Road','4','6584RO','Inglewood','070-458458','2983'),
(27,'Collins','DD','1990-05-10','F',2008,'Long Drive','804','8457DK','Eltham','079-234857','2513'),
(28,'Collins','C','1983-06-22','F',2008,'Old Main 28','10','1294QK','Midhurst','071-659599',Null),
(39,'Bishop','D','1986-10-29','M',2005,'Eaton Square','78','9629CD','Stratford','070-393435',Null),
(44,'Baker','E','1983-09-01','M',2010,'Lewis Street','23','4444LJ','Inglewood','070-368753','1124'),
(57,'Brown','M','1981-08-17','M',2007,'Edgecombe Way','16','4377CB','Stratford','070-473458','6409'),
(83,'Hope','PK','1976-11-11','M',2009,'Magdalene Road','16A','1812UP','Stratford','070-353548','1608'),
(94,'Miller','P','1993-05-14','M',2013,'High Street','33A','5746OP','Douglas','070-867564',Null),
(100,'Parmenter','P','1983-02-28','M',2012,'Haseltine Lane','80','1234KK','Stratford','070-494593','6524'),
(104,'Moorman','D','1990-05-10','F',2014,'Stout Street','65','9437AO','Eltham','079-987571','7060'),
(112,'Bailey','IP','1983-10-01','F',2014,'Vixen Road','8','6392LK','Plymouth','010-548745','1319');

#--Create the table committee_members and fill it--------------------

Create table committee_members
(
playerno   int   not null,
begin_date   date   not null,
end_date   date,
position   varchar(20),
primary key(playerno, begin_date)
);

Insert into committee_members values
(2,'2010-01-01','2012-12-31','Chairman'),
(2,'2014-01-01',Null,'General Member'),
(6,'2010-01-01','2010-12-31','Secretary'),
(6,'2011-01-01','2012-12-31','General Member'),
(6,'2012-01-01','2013-12-31','Treasurer'),
(6,'2013-01-01',Null,'Chairman'),
(8,'2010-01-01','2010-12-31','Treasurer'),
(8,'2011-01-01','2011-12-31','Secretary'),
(8,'2013-01-01','2013-12-31','General Member'),
(8,'2014-01-01',Null,'General Member'),
(27,'2010-01-01','2010-12-31','General Member'),
(27,'2011-01-01','2011-12-31','Treasurer'),
(27,'2013-01-01','2013-12-31','Treasurer'),
(57,'2012-01-01','2012-12-31','Secretary'),
(94,'2014-01-01',Null,'Treasurer'),
(112,'2012-01-01','2012-12-31','General Member'),
(112,'2014-01-01',Null,'Secretary');

#--Create the table matches and Fill it-------------------

Create table matches
(
matchno       int   not null   Primary Key,
teamno       int   not null   references teams(teamno),
playerno   int   not null   references players(playerno),
won       int,
lost       int
);

Insert into matches values
(1,1,6,3,1),
(2,1,6,2,3),
(3,1,6,3,0),
(4,1,44,3,2),
(5,1,83,0,3),
(6,1,2,1,3),
(7,1,57,3,0),
(8,1,8,0,3),
(9,2,27,3,2),
(10,2,104,3,2),
(11,2,112,2,3),
(12,2,112,1,3),
(13,2,8,0,3);

#--Create Table Penalties and Fill it-------------------------------------

create table Penalties
(
paymentno   int   not null   Primary Key,
playerno   int   not null   references players(playerno),
payment_Date   date   not null,
amount   decimal(10,2)   not null
);

Insert into Penalties values
(1,6,'2010-12-08',100.00),
(2,44,'2011-05-05',75.00),
(3,27,'2013-09-10',100.00),
(4,104,'2014-07-08',50.00),
(5,44,'2010-12-08',25.00),
(6,8,'2010-12-08' ,25.00),
(7,44,'2012-12-30',30.00),
(8,27,'2014-08-12',75.00);

#--Create Table Teams and Fill it----------------------------------------------

Create table teams
(
teamno   int   Primary Key   Not Null,
playerno   int   Not Null   references players(playerno),
division   varchar(6)
);

Insert into teams values
(1,6,'first'),
(2,27,'second');

/* ********************************************************************************************
End of loading the database
******************************************************************************************** */