Many human diseases are genetically transmitted (for example, hemophilia or Tay-Sachs disease). Here is a simple model for such a disease. The genotype aa is diseased and dies before it mates. The genotype Aa is a carrier but is not diseased. The genotype AA is not a carrier and is not diseased.

a. If two carriers mate, what are the probabilities that their offspring are of each of the three genotypes?

b. If the male offspring of two carriers is not diseased, what is the probability that he is a carrier?

c. Suppose that the non-diseased off spring of part (b) mates with a member of the population for whom no family history is available and who is thus assumed to have probability p of being a carrier ( p is a very small number). What are the probabilities that their first offspring has the genotypes AA, Aa, and aa?

d. Suppose that the first offspring of part (c) is not diseased. What is the probability that the father is a carrier in light of this evidence?

full explanation i dont understand thanks

With the COVID-19 crisis, a manager is concerned with handling the amount of customers arriving to his store. The manager assumes that the number of customers, X, arriving per hour has a Poisson distribution with mean rate of 10 customers per hour. Use this distribution for exercises!

1. Use R to generate the pmf of X.

2. Use R to plot the pmf of X.

3. Give the probability that at least 11 customers arrive to the store in a given hour.

4. There are 10 cards face down on a table and 3 of them are aces. If 5 of these cards are selected at random, what is the probability that at least 2 of them are aces?

5. Defects in a certain type of aluminum screen appear on the average of one in 150 square feet. If we assume the Poisson distribution, find the probability of at most one defect in 230 square feet.

According to Nielsen Media Research, the average number of hours of TV viewing by adults (18 and over) per week in the United States is 36.07 hours. Suppose the standard deviation is 8.6 hours and a random sample of 50 adults is taken. Appendix A Statistical Tables a. What is the probability that the sample average is more than 38 hours? b. What is the probability that the sample average is less than 38.5 hours? c. What is the probability that the sample average is less than 30 hours? If the sample average actually is less than 40 hours, what would it mean in terms of the Nielsen Media Research figures? d. Suppose the population standard deviation is unknown. If 66% of all sample means are greater than 35 hours and the population mean is still 36.07 hours, what is the value of the population standard deviation?

Problem 15-7 (Algorithmic)

Speedy Oil provides a single-server automobile oil change and lubrication service. Customers provide an arrival rate of 4.5 cars per hour. The service rate is 6 cars per hour. Assume that arrivals follow a Poisson probability distribution and that service times follow an exponential probability distribution.

1. What is the average number of cars in the system? If required, round your answer to two decimal places
   
   L =
   
2. What is the average time that a car waits for the oil and lubrication service to begin? If required, round your answer to two decimal places.
   
   W_q =  hours
   
3. What is the average time a car spends in the system? If required, round your answer to two decimal places.
   
   W =  hours
   
4. What is the probability that an arrival has to wait for service? If required, round your answer to two decimal places.
   
   P_w =

The number of customer arrivals at a bank's drive-up window in a 15-minute period is Poisson distributed with a mean of seven customer arrivals per 15-minute period. Define the random variable x to be the time (in minutes) between successive customer arrivals at the bank's drive-up window. (a) Write the formula for the exponential probability curve of x. (c) Find the probability that the time between arrivals is (Round your answers to 4 decimal places.) 1.Between one and two minutes. 2. Less than one minute 3. More than three minutes. 4. Between 1/2 and 3½ minutes. (d) Calculate μx, σ2x , and σx. (e) Find the probability that the time between arrivals falls within one standard deviation of the mean; within two standard deviations of the mean. (Round your answers to 4 decimal places.)

Suppose the length of textbooks in a library follows a bimodal distribution with a little right skewness (very mild). The mean of this distribution is 512 pages with a standard deviation of 390 pages.

For each of the following i) draw a picture. ii) label the picture with 2 axes (underneath). iii) label the shorthand for the new distribution. iv) Find the z-score. v) Find the answer.  

a1) What is the probability that a random sample of 36 textbooks has an average of 445.2 pages or less?

a2) What is the probability that a random sample of 49 textbooks has an average higher than 613.3 pages?

a3) What is the probability that a random sample of 30 textbooks has an average number of pages between 400 and 500?

a4) Do you think you could answer a1 - a4 with a sample of just 2 textbooks? Why or why not?

Approximately 85% of applicants get their G1 driver's license the first time they try the test. 80 applicants are scheduled to take the next test. The random variable X = The number of successful applicants during one session of administering the test.

a) Can this situation be approximated by a normal distribution? Explain. (2)

b) Using an approximation to the normal distribution, calculate the probability that exactly 65 applicants will pass the test (4)

c) Using the binomial distribution, calculate the probability that exactly 65 applicants will pass the test (2)

d) Using an approximation to the normal distribution, calculate the probability that more than 10 applicants will need to retake the test (4)

e) Can d) be answered using the binomial distribution? If so, which method (binomial distribution, approximation to the normal distribution) would you suggest someone use to find the answer? Explain. (2)

Worth 14 marks in total

In Assignment 2 you considered the following setting: Near-sightedness (myopia) afflicts roughly 10% of children at age 5.

Assume that the school has a total of 600 students.

Part A. In this school, what is the expected number of students to have myopia? What is the variance?

Part B. If we assume that the students in the school form an independent and random sample, then approximate the probability that at most 60 of the students in the 600 student school have myopia.

When calculating the approximate probability, please be sure to justify the approximation (i.e. check any conditions) and show all your work in the calculation.

Part C. If we assume that the students in the school form an independent and random sample, then what is the approximate probability that at least 55 and less than 65 of the students in the 600 student school have myopia? Again, show all necessary work in this calculation.

Airlines sometimes overbook fights. Suppose that for a plane with 50 seats, 53 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the fight. The probability mass function of Y appears in the accompanying table.

Y 48 49 50 51 52 53 P(Y=y) 0.28 ? 0.25 0.06 0.05 0.05

a. Given the missing value P(Y=49).

b. What is the probability that the fight will accommodate all ticketed passengers who show up?

c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight?

d. Find E(Y) and E(Y^2).

e. Find V(Y).

4). Commemorative coins are being struck at the local foundry. A gold blank (a solid gold disc with no markings on it) is inserted into a hydraulic press and the obverse design is pressed onto one side of the disc (this step fails with probability 0.15). The work is examined and if the obverse pressing is good, the coin is put into a second hydraulic press and the reverse design is imprinted (this step fails with probability 0.08). The completed coin is now examined and if of sufficient quality is passed on for finishing (cleaning, buffing, and so on). Twenty gold blanks are going to undergo pressing for this commemorative coin. Assume that all pressings are independent of each other.

4a: What are the mean and variance of the number of good coins manufactured?

4b: If the blanks cost $300 each and the labor to produce the finished coins costs $3,000, what is the probability that the production cost to make the 20 coins (labor and materials) can be recovered by selling the coins for $500 each?