The lifetime of a particular bulb is a random variable with an average of μ = 2000 hours and a standard deviation of σ = 200 hours.

(a) What is the probability that a bulb has a lifetime between 2000 and 2400 hours?

(b) What is the probability that a bulb will last less than 1470 hours?

(c) In the case that a bulb lasts for more than 2100 hours, the probability that it will last more than 2300 hours You calculate.

(d) the probability of a bulb lasting less than 2150 hours when it lasts for more than 2050 hours. You calculate.

17#12

Consider a system with one component that is subject to failure, and suppose that we have 100 copies of the component. Suppose further that the lifespan of each copy is an independent exponential random variable with mean 15 days, and that we replace the component with a new copy immediately when it fails.

(a) Approximate the probability that the system is still working after 1725 days.
Probability ≈

(b) Now, suppose that the time to replace the component is a random variable that is uniformly distributed over (0,0.5). Approximate the probability that the system is still working after 1950 days.
Probability ≈≈

One study found that 2% of American children have a peanut allergy. Showing your calculations…

a) If two independent American children are selected at random, what is the probability that neither has a peanut allergy?

b) If two independent American children are selected at random, what is the probability that both have a peanut allergy?

c) If two independent American children are selected at random, what is the probability that exactly one child has a peanut allergy?

d) In a classroom of 25 randomly selected and independent American children, what is the probability that no children have peanut allergies?

Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of all possible outcomes of a single iteration of a certain experiment. Also suppose that, for each C ∈ F, the probability that the outcome of this experiment is contained in C is P(C).
Consider events A, B ∈ F with P(A) + P(B) > 0. Suppose that the experiment is iterated indefinitely, with each iteration identical and independent of all the other iterations, until it results in an outcome that is an element of A ∪ B, after which it stops. What is the probability that this procedure results in an outcome that is an element of A? Do not use conditional probability to answer this question.

2. The median of an income distribution is $30,000. The mean income is $35,000 and the standard deviation is $25,000. For each part, if possible compute the quantity, if not possible give your reason.

a) Is the distribution symmetric? Why or why not?

b) The probability that a randomly selected income being less than $35,000

c) The probability that a randomly selected income being less than $30,000

d) The probability that the average of 4 randomly selected incomes being between $30,000 and $40,000

e) The probability that the average of 100 randomly selected incomes being between $30,000 and $40,000.

An investigation of past consumer surveys done by a company reveals that 2/3 of customers contacted respond to the survey. The marketing manager wants to do a new survey and plans to contact 198 customers.

(a) How many responses should the manager expect to receive?

(b) Give an approximation of the probability that 140 or more customers will respond.

(c) Give an approximation of the probability that 135 to 150 customers will respond.

(d) Give an approximation of the probability that 130 or less customers will respond.

(e) Compare your approximate answers with the exact probability values obtained on Excel.

2. (8 pts.) The Centers for Disease Control and Prevention reports that the rate of Chlamydia infections among American women ages 20 to 24 is 2791.5 per 100,000. Take a random sample of three American women in this age group. (a) What is the probability that all of them have a Chlamydia infection? (b) What is the probability that none of them has a Chlamydia infection? (c) What is the probability that at least one of them has a Chlamydia infection? (d) What is the probability that at most one has a Chlamydia infection?

Please show work and explain.

The average income tax refund for the 2009 tax year was ​$3109. Assume the refund per person follows the normal probability distribution with a standard deviation of ​$917. Complete parts a through d below.

a. What is the probability that a randomly selected tax return refund will be more than $2000?

b. What is the probability that a randomly selected tax return refund will be between $1500 and $2900?

c. What is the probability that a randomly selected tax return refund will be between $3400 and 4000​?

d. What refund amount represents the 35th percentile of tax​ returns?

Suppose systolic blood pressure of 16-year-old females is approximately normally distributed with a mean of 119 mmHg and a variance of 398.00 mmHg. If a random sample of 16 girls were selected from the population, find the following probabilities:

a) The mean systolic blood pressure will be below 116 mmHg.

probability = b) The mean systolic blood pressure will be above 123 mmHg.

probability = c) The mean systolic blood pressure will be between 109 and 123 mmHg.

probability = d) The mean systolic blood pressure will be between 105 and 113 mmHg. probability =

Suppose systolic blood pressure of 17-year-old females is approximately normally distributed with a mean of 113 mmHg and a standard deviation of 24.71 mmHg. If a random sample of 22 girls were selected from the population, find the following probabilities:

a) The mean systolic blood pressure will be below 109 mmHg. probability =

b) The mean systolic blood pressure will be above 118 mmHg. probability =

c) The mean systolic blood pressure will be between 106 and 125 mmHg. probability =

d) The mean systolic blood pressure will be between 106 and 113 mmHg. probability =