In a recent school year in the state of Washington, there were 319,000 high school students. Of these, 154,000 were girls and 165,000 were boys. Among the girls, 41,100 dropped out of school, and among the boys, 10,500 dropped out. A student is chosen at random. Round the answers to four decimal places.

(a) What is the probability that the student is female?

(b) What is the probability that the student dropped out?

(c) What is the probability that the student is female and dropped out?

(d) Given that the student is female, what is the probability that she dropped out?

(e) Given that the student dropped out, what is the probability that the student is female?

Pavement requires aggregate of specific size. Aggregates for a highway pavement are extracted from a gravel pit. Based on experience with this area, it is known that the probability of good-quality aggregate is only 80%. In order to weed out poor-quality aggregate, engineers on site use a quick test, which is not entirely reliable. The probability that a good-quality aggregate passes is 90%, whereas the probability that a poor-quality aggregate will pass is 20%.
⦁   What is the probability that a random sample of gravel from the pit passes the test?
⦁   What is the probability that a passing sample is poor-quality?
⦁   What is the overall accuracy of this test?

In a group of students, there are 2 out of 18 that are left-handed.

a. Assuming a low-informative prior probability distribution, find the posterior distribution of left-handed students in the population. Summarize your results with an estimation of the mean, median, mode, and a 95% credible interval. Plot your posterior probability distribution.

b. According to the literature, 5 to 20% of people are left-handed. Take this information into account in your prior probability and calculate a new posterior probability distribution. Summarize your results with an estimation of the mean, median, mode, and a 95% credible interval. Plot your posterior probability distribution.

Please use R studio, Thank you.

2. The probability of a student passing statistics is known to be 0.41; and the probability of a student passing chemistry is known to be 0.55. If the probability of passing both is known to be 0.35, calculate:

(a) the probability of passing at least one of statistics and chemistry

(b) the probability of a student passing chemistry, given that they passed statistics

(c) Are passing chemistry and statistics independent? Justify

(d) (harder) a group of 33 randomly selected students attend a special seminar on study skills. Of these 33, only 7 fail both. State a sensible null hypothesis, test it, and interpret.

A laptop assembly is subjected to a final functional test. Suppose that defects occur at random in these assemblies, and that defects occur according to a Poisson distribution with parameter λ= 0.03.

1. What is the probability that an assembly will have exactly one defect?
2. What is the probability that an assembly will have one or more defects?
3. What is the probability that an assembly will have no more than three defects?
4. What is the probability that an assembly will have more than two defects?
5. Suppose that you improve the process so that the occurrence rate of defects is cut in half to λ = 0.015. What effect does this have on the probability that an assembly will have one or more defects?

By rewriting the formula for the multiplication​ rule, you can write a formula for finding conditional probabilities.

The conditional probability of event B​ occurring, given that event A has​ occurred, is Upper P left parenthesis Upper B vertical line Upper A right parenthesis equals StartFraction Upper P left parenthesis Upper A and Upper B right parenthesis Over Upper P left parenthesis Upper A right parenthesis EndFraction . Use the information below to find the probability that a flight departed on time given that it arrives on time.

The probability that an airplane flight departs on time is 0.91.

The probability that a flight arrives on time is 0.86.

The probability that a flight departs and arrives on time is 0.82.

The probability that a flight departed on time given that it arrives on time is 2 nothing. ​(Round to the nearest thousandth as​ needed.)

Here is the information for a class on campus. We need to randomly select two students to interview. (We don't want to interview the same student twice.) Class Frequency Freshman 21 Sophomore 14 Junior 9 Senior 6

(a) Are the two student selections going to be independent?

(b) What is the probability that we choose a Freshman and then a Senior?

(c) What is the probability that we choose a Sophomore and then another Sophomore?

(d) What is the probability that we choose a Freshman and a Senior in any order?

(e) What is the probability that both students are at the same class level?

(f) What is the probability that both students are not at the same class level?

(g) We decide we need to interview more students. We decide to interview a total of 6 randomly chosen students. What is the probability that all the students chosen are freshmen?

The weight of hamsters is normally distributed with mean 63.5 g and standard deviation 12.2 g.

A. What is the probability that a randomly selected hamster weighs less than 58.5 g? In order to receive full credit, sketch the density curve, fill in the area and give the appropriate probability notation. (5 points)

B. What is the probability that a randomly selected hamster weighs more than 70.7 g? In order to receive full credit, sketch the density curve, fill in the area and give the appropriate probability notation. (5 points)

C. What is the probability that a randomly selected hamster weighs between 65 g and 75 g? In order to receive full credit, sketch the density curve, fill in the area and give the appropriate probability notation. (10 points)

). A farmer knows from experience that his wheat harvest  Q (in        

                     bushels) has the probability distribution given by:

                                         Q = 150  with probability  0.30 ,

                                              = 170    with probability  0.45,

                                               =200    with probability  0.25;

                      note that the probabilities add up to 1 as they are required by a probability   

                      distribution. Suppose the demand function he faces in the market place is given by:

                                              p  =  320  -  0.5  Q

                       where  p =  price in dollars per bushel. Let  R  = total revenue = p x Q. [Note: You may find it

                      convenient to first derive the probability distribution of  R.]

Find E ( R ).

                      Also, define his Profits as :

                                            Profits =  R - C  ,

                     where the total cost  C  (in dollars) is a function of  Q  given by

                                            C =  150  -   10 Q   + 2 Q².

Find  E(Profits).

A set of final grades in an introductory probability class in section A is normal distributed with mean 70 and standard deviation of 10 : G1 ~ N (70,100). a) what is probability that grade of randomly chosen student is 77 or less? b)what is probability that average grade of 9 randomly chosen students is 77 or less? c) only 2% of the students scored higher that what grade? d) A set of final grades in an introductory probability class in section B is normally distributed with mean 75 and unknown standard deviation σ : G2 ~ N (75, σ2). If probability that G2 ≤ 80 is 0.8159, I)what is the value of σ ? II) what is the probability that randomly chosen student from class A scores more than randomly chosen student from class B ?