1. You are given Pr(A) = 0.3, Pr(B) = 0.5 and Pr(A ∪ B) = 0.6. (a) WhatisPr(A∩B)?

(b) What is Pr(notA)?

2. A bag contains 30 balls of the same size. Each ball is either red or green.

The probability of choosing a green ball is 0.7. How many of the balls are red?

3. Given nPr =42 and nCr =7 find the value of r?

4. We have Pr(A|B) = 0.3 and Pr(B) = 0.45. What is Pr(A ∩ B)?

5. How many 3 digit numbers can be created from the set {1, 2, 3, 4, 5, 6} if the digits cannot be repeated?

6. What is the probability of choosing 10 cards at random from a standard deck of 52 cards and getting exactly 2 queens?

7. What is the variance of the numbers 3, 7, and 8?

8. We have a discrete probability distribution where P r(X = 1) = 0.2,

Pr(X = 2) = 0.3 and Pr(X = 3) = 0.5. Determine μ.

9. We have a discrete probability distribution where P r(X = 1) = 0.6,

Pr(X=3)=0.1,Pr(X=5)=0.3andμ=2.4. Determineσ2.

10. We have a binomial distribution where the probability of success is 0.4 and the number of trials is 3,000. What is the variance of the distribution?

HELP 1-10!!!!!!!!

{Exercise 4.19 (Algorithmic)} The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States: 2002). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here. Participants (millions)             Activity                                     Male                                     Female             Bicycle riding                                     20.7                                     22.5             Camping                                     24.1                                     25.8             Exercise walking                                     30.2                                     59.2             Exercising with equipment                                           18.9                                     25.9             Swimming                                     24.9                                     32.9                                     For a randomly selected female, estimate the probability of participation in each of the sports activities (to 2 decimals). Note that the probabilities do not sum to 1 because of participation in more than one sports activity. Bicycle riding Camping Exercise walking Exercising with equipment Swimming For a randomly selected male, estimate the probability of participation in each of the sports activities (to 2 decimals). Note that the probabilities do not sum to 1 because of participation in more than one sports activity. Bicycle riding Camping Exercise walking Exercising with equipment Swimming For a randomly selected person, what is the probability the person participates in exercise walking (to 2 decimals)? Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman (to 2 decimals)? What is the probability the walker is a man (to 2 decimals)?

When returns from a project can be assumed to be normally distributed, such as those shown in Figure 13-6 (represented by a symmetrical, bell-shaped curve), the areas under the curve can be determined from statistical tables based on standard deviations. For example, 68.26 percent of the distribution will fall within one standard deviation of the expected value ( D¯¯¯D¯ ± 1σ). Similarly, 95.44 percent will fall within two standard deviations ( D¯¯¯D¯  ± 2σ), and so on. An abbreviated table of areas under the normal curve is shown next.
  

Assume Project A has an expected value of $20,000 and a standard deviation (σ) of $4,000.

a. What is the probability that the outcome will be between $18,000 and $22,000? (Round your answer to 4 decimal places.)
  

  
b. What is the probability that the outcome will be between $16,000 and $24,000? (Round your answer to 4 decimal places.)
  

  
c. What is the probability that the outcome will be at least $12,000? (Round your answer to 4 decimal places.)
  

  
d. What is the probability that the outcome will be less than $33,830? (Round your answer to 4 decimal places.)
  

  
e. What is the probability that the outcome will be less than $18,000 or greater than $22,000? (Round your answer to 4 decimal places.)
  

At Perry’s Pumpkin Patch there are 50 pumpkins to choose from in four different colors. Five pumpkins are yellow, 12 are green, 6 are white and the rest are orange pumpkins. Answer the following questions. PLEASE SHOW STEPS/CALCULATIONS

a) You will randomly choose 8 pumpkins to take home with you. What is the probability that you have chosen at least 2 orange pumpkins? What distribution, parameter(s), and support are you using?

b) Given that you have chosen at least 2 orange pumpkins, what is the probability that you have chosen fewer than 5 orange pumpkins?

c) What is the expected value and standard deviation of the number of white pumpkins you have chosen?

d) Over on another field there are 300 pumpkins, 20 of which are white. You are going to randomly choose 10 pumpkins from this larger field. What is the probability that you have chosen at most one white pumpkin from the second field?

e) Is there an approximation that can be used on the second field to find the probability that you have chosen at most one white pumpkin? Justify why you can or cannot use an approximation and state its distribution, parameter(s), and support.

f) Find the approximate probability that you have chosen at most one white pumpkin from the second field.

Mr. Keller filled out a bracket for the NCAA National Tournament. Based on his knowledge of college basketball, he has a 0.5 probability of guessing any one game correctly.
What is the probability Mr. Keller will pick all 32 of the first round games correctly? Incorrect
What is the probability Mr. Keller will pick exactly 8 games correctly in the first round? Incorrect
What is the probability Mr. Keller will pick exactly 20 games incorrectly in the first round?

A high school baseball player has a 0.261 batting average. In one game, he gets 9 at bats. What is the probability he will get at least 7 hits in the game?

Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 6.2-in and a standard deviation of 0.9-in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 1.6% or largest 1.6%.
What is the minimum head breadth that will fit the clientele?
min =
What is the maximum head breadth that will fit the clientele?
max =
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

(Normal approximation to Binomial) Suppose that each student has probability p of correctly answering a question chosen at random from a large inventory of questions. The correctness of answer to a question is independent of the correctness of answers to other questions. An exam consisting of n questions can be considered as a simple random sample of size n from the population of all questions. Therefore, the number of correct answers X in an exam of n questions is a Binomial(n, p) random variable. This is true for every student writing the test. However, the success probability p varies from student to student.

Assume that each question is worth 1 point, that is, the maximum marks is n. In the following questions, use Normal approximation to compute the Binomial probabilities. Use continuity correction,

1. Roxi is a good student for whom p = 0.85. Compute the probability that Roxi will score 90% or better on a 100 question exam.
2. If the exam contains 200 questions, what is the probability that Roxi will score 90% or better? Note that 90% of 200 is 180.
3. Zulekha is a weaker student for whom p = 0.65. Suppose the passing grade for the exam is 70%. What is the probability that Zulekha will pass a 200 questions exam?
4. Suppose Zulekha is given a choice to write a 100 questions exam instead of 200 questions exam. Should she write the shorter exam? Show your calculations to justify your answer.

Question 1
Given an example on any one of the following sampling techniques (one example on only one technique).

 Simple random sampling

 Stratified random sampling

 Cluster sampling
Question 2
What is the difference between sampling error and non-sampling error? Give an example on each part.

Question 3

a. Find P(A1)

b. Find P(A1 and B2)

c. Find P(A1|B2)

d. Find P(A2 or B1)

e. Find (B1 and B2)

Question 4
X the number of TVs a family owns in Doha. The table below shows the probability distribution.

a. What is the probability that a family owns 3 TVs or less?

b. What is the probability that a family own exactly 2 TVs?

c. Find the standard deviation of the discrete random variable X

Question 5
A sales man finds that on average 0.15 of the TVs he sells are rejected each month because they are either too big or too small in size. The sales man has 20 TVs to sell each month.
a. What is the probability that at least 3 TVs be rejected.

b. What is the probability that 4 TVs are rejected.

2. Suppose that you are waiting for a friend to call you and that the time you wait in minutes has an exponential distribution with parameter λ = 0.1. (a) What is the expectation of your waiting time? (b) What is the probability that you will wait longer than 10 minutes? (c) What is the probability that you will wait less than 5 minutes? (d) Suppose that after 5 minutes you are still waiting for the call. What is the distribution of your additional waiting time? In this case, what is the probability that your total waiting time is longer than 15 minutes? (e) Suppose now that the time you wait in minutes for the call has a U(0, 20) distribution. What is the expectation of your waiting time? If after 5 minutes you are still waiting for the call, what is the distribution of your additional waiting time?

3. The arrival times of workers at a factory first-aid room satisfy a Poisson process with an average of 1.8 per hour. (a) What is the value of the parameter λ of the Poisson process? (b) What is the expectation of the time between two arrivals at the first-aid room? (c) What is the probability that there is at least 1 hour between two arrivals at the firstaid room? (d) What is the distribution of the number of workers visiting the first-aid room during a 4-hour period? (e) What is the probability that at least four workers visit the first-aid room during a 4- hour period?

Engineering system of type k-out-of-n is operational if at least k out of n components are operational. Otherwise, the system fails. Suppose that a k-out-of-n system consists of n identical and independent elements for which the lifetime has Weibull distribution with parameters r and λ. More precisely, if T is a lifetime of a component, P(T ≥ t) = e−λtr, t ≥ 0. Time t is in units of months, and consequently, rate parameter λ is in units (month)−1. Parameter r is dimensionless. Assume that n = 8,k = 4, r = 3/2 and λ = 1/10. (a) Find the probability that a k-out-of-n system is still operational when checked at time t = 3. (b) At the check up at time t = 3 the system was found operational. What is the probability that at that time exactly 5 components were operational? Hint: For each component the probability of the system working at time t is p = e−0.1 t3/2. The probability that a k-out-of-n system is operational corresponds to the tail probability of binomial distribution: IP(X ≥ k), where X is the number of components working. You can do exact binomial calculations or use binocdf in Octave/MATLAB (or dbinom in R, or scipy.stats.binom.cdf in Python when scipy is imported). Be careful with ≤ and <, because of the discrete nature of binomial distribution. Part (b) is straightforward Bayes formula.

The owner of a small deli is trying to decide whether to discontinue selling magazines. He suspects that only 10% of his customers buy a magazine and he thinks that he might be able to use the display space to sell something more profitable. Before making a final decision, he decides that for one day he will keep track of the number of customers that buy a magazine.

(a) Explain why this is a binomial experiment.



(b) Assuming his suspicion that 10% of his customers buy a magazine is correct, what is the probability that exactly 6 out of the first 10 customers buy a magazine? Give your answer as a decimal number rounded to two digits.



(c) What is the expected number of customers from this sample that will buy a magazine?