A and B play the following game: A writes down either number 1 or number 2, and B must guess which one. If the number that A has written down is i and B has guessed correctly, B receives i units from A. If B makes a wrong guess, B pays i unit to A. If B randomizes his decision by guessing 1 with probability p and 2 with probability 1 - p, determine his expected gain if

(a) A has written down number 1 and

(b) A has written down number 2.

What value of p maximizes the minimum possible value of B's expected gain, and what is this maximin value? (Note that B's expected gain depends not only on p, but also on what A does.)

Consider now player A. Suppose that she also randomizes her decision, writing down number 1 with probability q. What is A's expected loss if

( c) B chooses number 1 and

( d) B chooses number 2?

What value of q minimizes A's maximum expected loss? Show that the minimum of A's maximum expected loss is equal to the maximum of B's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player B.

USA Today reported that approximately 25% of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let x = number of prisoners out of five on parole who become repeat offenders.

(a) Find the probability that one or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


How does this number relate to the probability that none of the parolees will be repeat offenders?

These probabilities are not related to each other.This is twice the probability of no repeat offenders.     This is five times the probability of no repeat offenders.These probabilities are the same.This is the complement of the probability of no repeat offenders.

(b) Find the probability that two or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


(c) Find the probability that four or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


(d) Compute μ, the expected number of repeat offenders out of five. (Round your answer to three decimal places.)
μ =  prisoners

(e) Compute σ, the standard deviation of the number of repeat offenders out of five. (Round your answer to two decimal places.)
σ =  prisoners

The college student senate is sponsoring a spring break Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Center for the Homeless. A local travel agency donated the cruise, valued at $2000. The students sold 2034 raffle tickets at $5 per ticket.

(a) Kevin bought nineteen tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? (Round your answer to five decimal places.)


What is the probability that Kevin will not win the cruise? (Round your answer to five decimal places.)


(b) Expected earnings can be found by multiplying the value of the cruise by the probability that Kevin will win. What are Kevin's expected earnings? (Round your answer to two decimal places.)
$  

Is this more or less than the amount Kevin paid for the nineteen tickets?
---Select--- more less

How much did Kevin effectively contribute to the Samaritan Center for the Homeless? (Round your answer to two decimal places.)
$

USA Today reported that approximately 25% of all state prison inmates released on parole become repeat offenders while on parole. Suppose the parole board is examining five prisoners up for parole. Let x = number of prisoners out of five on parole who become repeat offenders.

(a) Find the probability that one or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


How does this number relate to the probability that none of the parolees will be repeat offenders?

These probabilities are not related to each other.This is twice the probability of no repeat offenders.     This is five times the probability of no repeat offenders.These probabilities are the same.This is the complement of the probability of no repeat offenders.

(b) Find the probability that two or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


(c) Find the probability that four or more of the five parolees will be repeat offenders. (Round your answer to three decimal places.)


(d) Compute μ, the expected number of repeat offenders out of five. (Round your answer to three decimal places.)
μ =  prisoners

(e) Compute σ, the standard deviation of the number of repeat offenders out of five. (Round your answer to two decimal places.)
σ =  prisoners

The college student senate is sponsoring a spring break Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Center for the Homeless. A local travel agency donated the cruise, valued at $2000. The students sold 2034 raffle tickets at $5 per ticket.

(a) Kevin bought nineteen tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? (Round your answer to five decimal places.)


What is the probability that Kevin will not win the cruise? (Round your answer to five decimal places.)


(b) Expected earnings can be found by multiplying the value of the cruise by the probability that Kevin will win. What are Kevin's expected earnings? (Round your answer to two decimal places.)
$  

Is this more or less than the amount Kevin paid for the nineteen tickets?
---Select--- more less

How much did Kevin effectively contribute to the Samaritan Center for the Homeless? (Round your answer to two decimal places.)
$

Consider a firm that is introducing a new product. The firm identified 300 potential customers whose probability of purchasing the product depends on age and gender as follows:

Using the spreadsheet of customer data provided, estimate the average number of product demand in each city: New York, Chicago, Los Angeles, and Seattle.

Color blindness is the decreased ability to see colors and clearly distinguish different colors of the visible spectrum. It is known that color blindness affects 11% of the European population. A sample of 12 subjects is taken for research purposes by a team of optometrists.

What is the probability that one subject is color blind?

What is the probability that more than one subject is color blind?

What is the probability that at most 3 subjects are color blind?

What is the probability that the number of color blind subjects is between 2 and 4 (both included)?

What is the probability that at least 10 subjects are not color blind?

(a) Briefly discuss the binomial probability distribution.

(b) A coin is flipped 12 times: what is the probability of getting:

i. no heads; and)

ii. no more than 3 heads?

Over the past 10 years two golfers have had an ongoing battle as to who the better golfer is. Curtley Weird has won 120 of their 200 matches, while Dave Chilly has won 70 with 10 of them ending in ties. Because Dave is going overseas they decide to play a tournament of five matches to establish once and for all who the better player is.

Find the probabilities that:

(a) Dave wins at least three of the matches;

(b) Curtley wins no more than two games; and

(c) all of the games end in a tie.

(a) Discuss probability, independence and mutual exclusivity, giving examples to illustrate your answer.

i. How many ways are there of choosing a committee of three people from a club of ten?

ii. How many ways are there of selecting from those three people a president, secretary and treasurer?

iii. Illustrate your answer to the second part of the question with a tree diagram.

An ice-cream vendor on the beachfront knows from long experience that the average rate of ice-cream sales is 12 per hour. If, with two hours to go at work, she finds herself with only five ice-creams in stock, what are the probabilities that

(a) she runs out before the end of the day;

(b) she sells exactly what she has in stock by the end of the day without any excess demand after she sells the last one; and

(c) she doesn't sell any?

A company applying for medical aid cover counts that 70 of its 140 male employees smoke. Of the 100 female employees, 20 smoke. What is the probability that an employee chosen at random

(a) is female and smokes; (2)

b) does not smoke; and

(c) is male or smokes?

In a true or false assignment of six questions you are obliged to get at least four correct to pass. If you guess the answers to the questions, what are the probabilities that:

a) you pass; (4)

(b) you get at least 50% of the answers correct; and

(c) you get no more than two correct? (3)

onist claims that he gets 10 calls every five minutes. To demonstrate this to his boss he makes a tape lasting five minutes. What are the probabilities that he gets:

(a) no calls in the five minutes; (2)

(b) less than three calls; and (5)

(c) exactly 10 calls? (

Assume that matric marks are standardised to have a mean of 52% and a standard deviation of 16% (and assume that they have a normal distribution). In a class of 100 students estimate how many of them:

(a) pass (in other words get more than 33,3%);

(b) get A's (more than 80%); and

(c) get B's (between 70% and 80%).

As manager of a company you know that the distribution of completion times for an assembly operation is a normal distribution with a mean of 120 seconds and a standard deviation of 20 seconds. If you have to award bonuses to the top 10% of your workers what time would you use as a cut-off time? [6]

Statistical data related to college football was gathered over the following.  

State the Highest Level of Measurement of each.

a) Time of kickoff (11:00 am, 2:00 pm, 7:00 pm).

b) Number of fans attending the game.

c) Temperature at kickoff.

d) Types of beverages sold at the stadium (water, pop, gatorade, etc).

e) Rating of stadium experience (poor, average, excellent).

f) Cost of ticket.

in PYTHON - Kyle Lowry is trying to calculate his career statistics thus far in his illustrious career. Write a program that prompts the user for Lowry’s average points per game (ppg) in EACH of his 14 seasons in the NBA. The program will display the number of seasons in which he averaged 15 ppg or more, his career average, and his best season (SEASON # in which he had his highest ppg).

Let H ≡ the number of punches landed by a boxer on her opponent. If H ~ BIN (20, 0.7), what is the probability that the average number of hits by a random sample of 55 boxers is between 13.7 and 14.5?