discrete probability distributions

There are 37 different processors on the motherboard of a controller. 6 of the processors are faulty. It is known that there are one or more faults on the motherboard. In an attempt to locate the error, 7 random processors are selected for testing.

tasks

a) Determine the expected number of defective processors. Round your answer to 2 decimal places.

b) Determine the variance of the number of defective processors. Round your answer to 4 decimal places.

c) Determine the standard deviation of the number of defective processors. Round your answer to 2 decimal places.

d) What is the probability that there are at least 2 faulty processors? Round your answer to 4 decimal places.

e) What is the probability that there are exactly 1 faulty processors if there are a maximum of 2 faulty processors?  Round your answer to 4 decimal places.

TV sets: According to the Nielsen Company, the mean number of TV sets in a U.S. household in 2013 was 2.24 . Assume the standard deviation is 1.2 . A sample of 95 households is drawn. Use the Cumulative Normal Distribution Table if needed.

What is the probability that the sample mean number of TV sets is greater than 2? Round your answer to four decimal places.

What is the probability that the sample mean number of TV sets is between 2.5 and 3? Round your answer to four decimal places.

Find the 80th percentile of the sample mean. Round your answer to two decimal places.

Would it be unusual for the sample mean to be less than 2? Round your answer to four decimal places.

In a couple's family history, male children dominate the offspring. This couple seeks to break the tradition and decides to have children until they have a desired number of girls. Let us suppose that there is no factor that indicates that the number of men in your family has biological causes and that the distribution of women and men in your family has been a mere coincidence.

a)Find the probability that the family will have four children if they have decided to have children until the first girl appears.

b) What is the probability that the couple will have at most 4 children if they only expect to have a girl?

c) Find the probability that the family will have six children if they have decided to have three girls.

d) What is the probability that the couple will have at most six children if they want to have three girls?

e)How many children does this family expect to have? (calculate the children expected by each family for the situation in part a), part d) and then generalize for a number r of girls)

f)How many children (boys) does this family expect to have? (calculate the boys that each family expects for the situation in part a), part c) and then generalize for a number r of girls)

The probability of winning the Powerball jackpot on a single given play is 1/175,223,510. Suppose the powerball jackpot becomes large, and many people play during one particular week. In fact, 180 million tickets are sold that week. Assuming all the tickets are independent of one another, then the number of tickets should be binomially distributed. The values of the parameters n and p in this binomial distribution are:

n=

p=

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold.

______

If X = the number of winning tickets sold, find the mean and standard deviation of the random variable X.

Mean of X =  

Standard deviation of X =  

On the other hand, since the "times" between winning tickets should be independent of one another, the number of winning tickets per week could reasonably be modeled by a Poisson distribution.

The value of the parameter lambda in this Poisson distribution would be _____

The standard deviation of the number of tickets sold in a week (using the Poisson model) is _________

What does the Poisson model predict is the probability of having one or more winning tickets sold? _____

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold. _________

At his workplace, the ?rst thing Oscar does every morning is to go to the supply room and pick up one, two, or three pens with equal probability 1/3. If he picks up three pens, he does not return to the supply room again that day. If he picks up one or two pens, he will make one additional trip to the supply room, where he again will pick up one, two, or three pens with equal probability 1/3. (The number of pens taken in one trip will not a?ect the number of pens taken in any other trip.) Calculate the following:

(a) The probability that Oscar gets a total of three pens on any particular day.

(b) The conditional probability that he visited the supply room twice on a given day, given that it is a day in which he got a total of three pens.

(c) E[N] and E[N |C], where E[N] is the unconditional expectation of N, the total number of pens Oscar gets on any given day, and E[N |C] is the conditional expectation of N given the event C = {N > 3}.

(d) ?N |C, the conditional standard deviation of the total number of pens Oscar gets on a particular day, where N and C are as in part (c).

(e) The probability that he gets more than three pens on each of the next 16 days. (f) The conditional standard deviation of the total number of pens he gets in the next 16 days given that he gets more than three pens on each of those days.

Information for Problems 1 – 9: The diameter of piston rings produced for automobile engines is known to be normally distributed with a population standard deviation,std, equal to 0.100 millimeters. The last ten piston rings produced by a particular manufacturer have the following diameters (in millimeters): 74.036, 74.432, 74.212, 74.071, 73.968, 74.231, 73.899, 74.035, 74.079 and 73.995

1.     If you were to calculate a confidence interval for the mean diameter of piston rings, would you use a Z-Interval or a T-Interval?

2.     If you were to calculate a two-sided 94% confidence interval on the mean diameter of piston rings, what would be the value of the |a_a/2| or |t_a/2| that you would use? Round to 3 decimal places.

3.     What is the two-sided 94% confidence interval on the mean diameter of piston rings given the data you have?

4.    Where does u fall in this interval? Where does x(line above) fall in this interval?

5.     What is the margin of error for this 94% confidence interval? In other words, what is the amount that you are adding and subtracting from x(line above)?

6.     If the manufacturer wants the error on this 94% confidence interval to be within 0.03 of the true mean, what sample size is needed?

7.     If the confidence level is decreased, would the width of the confidence interval increase or decrease? What variable in the formula is the cause of the increase or decrease?

8.     If you were to calculate a one-sided 94% confidence interval on the mean diameter of piston rings, what would be the value of the |z_a| or |t_a| that you would use?  

9.     Calculate the one-sided 94% Upper Bound on the mean diameter of piston rings.

Information for Problems 10 – 18: A civil engineer tests the compressive strength of 12 batches of concrete.   It is known that the compressive strength of concrete is normally distributed; however, the population standard deviation is unknown. The compressive strength is obtained from the following 12 readings (in psi): 2216, 2225, 2318, 2237, 2301, 2255, 2249, 2281, 2275, 2204, 2263, 2295

10. If you were to calculate a confidence interval for the mean compressive strength of the concrete, would you use a Z-Interval or a T-Interval?

11. If you were to calculate a two-sided 92% confidence interval for the mean compressive strength of the concrete, what would be the value of the |z_a/2| or |t_a/2| that you would use?  

12. What is the two-sided 92% confidence interval for the mean compressive strength of the concrete?

13. Where does u fall in this interval? Where does x(line above) fall in this interval?

14. What is the margin of error for this 92% confidence interval?

15. If the civil engineer wants the error on this 92% confidence interval to be within 15 psi of the true mean, what sample size is needed?

16. If you were to calculate a one-sided 92% confidence interval for the mean compressive strength of the concrete, what would be the value of the |z_a| or |t_a| that you would use?  

17. Calculate the one-sided 92% Lower Bound for the mean compressive strength of the concrete.

18. Consider the two-sided confidence interval again. If the sample size were increased, but the mean and standard deviation stayed the same, does the precision increase or decrease?

Information for Problems 19 – 24: An article in Knee Surgery, Sports Traumatology, Arthroscopy showed that of the 25 tears that were reported to be located between 3 mm and 6 mm from the meniscus, only 15 healed after surgery.

19. What is the sample proportion, p , of such tears that heal?

20. If you were to calculate a two-sided 89% confidence interval on the proportion of tears that heal, would you use |z_a/2| or |t_a/2| and what is the value?  

21. Calculate a two-sided 89% confidence interval on the proportion of such tears that will heal.   

22. If the researchers want the error on the 89% confidence interval to be within 0.10 of the true proportion of tears that heal after surgery, what sample size is needed? Use the estimated value, p , to do your calculations.

23. Repeat #22, but this time assume you do not have an estimated value for p.

24. Calculate a one-sided upper 89% confidence bound on the proportion of such tears that will heal.

Information for Problems 25 – 30: Suppose city police calculate the 95% confidence interval on the proportion of calls for drug overdoses that end in fatalities to be (0.59259, 0.92741).

25. What must be the value of p, the point estimate for the proportion of calls for drug overdoses that end in fatalities?

26. What must be the value of q?

27. What is the critical score that would be used to calculate the 95% confidence interval?

28. When can you use a T-score to calculate a confidence interval for a proportion?

29. If a sample size of 25 is used to calculate this confidence interval, how many fatalities must have occurred?

30. If the police want to be more precise in their estimate of the proportion of fatalities that result from drug overdoses, they should _____________ (increase/decrease) the level of confidence.

This is math probability questions.

Please write the solution & answer on the paper and upload a picture of it if possible.

(Don'y use R plz)

1. One fair coin and two unfair coins where heads is 5 times as likely as tails are put into a bag. One coin is drawn at random and then flipped twice. If at least one of the flips was tails, what is the probability an unfair coin was flipped?

2. There are 4 quarters and 3 dimes and 1 nickel in a coin pouch. Two coins are selected at random. Find the probability density function of X, if X assigns the total value of the coins to each outcome. (You might want to draw a tree here.)

3. An unfair coin with heads five times as likely as tails is flipped four times. If X is the number of heads flipped, find the probability density function for X (Do not simplify your answers.!)

4. Balls are numbered 1 through 100 (including 1 and 100) in an urn. 3 are drawn simultaneously and at random. If X is two times the number of even balls drawn minus 3 times the number of odd balls drawn, find the probability density function for X.

Thank you.

(Problem 6)     In a popular day care center, the probability that a child will play with the computer is 0.45; the probability that he or she will play dress-up is 0.27; play with blocks, 0.18; and paint, 0.1. a) Construct the probability distribution for this discrete random variable.

b)         What is the Probability a child plays with a computer or paint

(Problem 7)     The county highway department recorded the following probabilities for the number of accidents per day on a certain freeway for one month. The number of accidents per day and their corresponding probabilities are shown. Find the mean, variance, and standard deviation.

Can you make sure I answered it correctly

Given:

Mean: 1.3

Variance= -1.66

SD= 1.29

A recent survey reported that 63% of​ 18- to​ 29-year-olds in a certain country own tablets. Using the binomial​ distribution, complete parts​ (a) through​ (e) below.

?: .52 ?=6

a. What is the probability that in the next six​ 18- to​ 29-year-olds surveyed, four will own a​ tablet?

The probability is ____

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

b. What is the probability that in the next six​ 18- to​ 29-year-olds surveyed, all six will own a​ tablet?

The probability is ____

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

c. What is the probability that in the next six​ 18- to​ 29-year-olds surveyed, at least four will own a​ tablet?

The probability is ____

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

d. What are the mean and standard deviation of the number of​ 18- to​29-year-olds who will own a tablet in a survey of​ six?

The mean number of​ 18- to​ 29-year-olds who own tablets out of six surveyed is _____

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

The standard deviation of the number of​ 18- to​ 29-year-olds who own tablets out of six surveyed is

nothing.

​(Type an integer or a decimal. Round to four decimal places as​ needed.)

1. Suppose the number of messages arriving on a communication line seems to be describable by a Poisson Random Variable with an average arrival rate of 10 messages/second. Determine the following

a. The probability that no messages arrive in a one second period.

b. The probability that 1 message arrives in a one second period.

c. The probability that 2 messages arrives in a one second period.

d. The probability that 3 messages arrives in a one second period.

e.The probability that 10 messages arrive in a one second period.

f. The probability that 4or more messages arrive in a one second period