Figure 1 shows the schematic diagram of a mechanical system that consists of two subsystems in parallel. The first subsystem consists of two components A and B in series, both of which have the same 0.80 probability of working. The second subsystem consists of three components C, D, and E in series, all of which have the same 0.90 probability of working. It is assumed that all components of the system are independent. (a) What is the probability that subsystem A-B works? (5 points) (b) What is the probability that subsystem C-D-E works? (5 points) (c) What is the probability that the whole system works? Hint: Remember that the probability that the system works = 1 – probability that the system doesn’t work. (7 points) (d) What is the probability that A is not working, given that the system works? Hint: Does it matter whether B works or not? (8 points)

Conduct your own experiment by rolling a standard die.

(a) List the possible outcomes. (Enter your answers as a comma-separated list.)

(b) Perform the experiment 36 times. Make a table to record your results.

This answer has not been graded yet.

(c) Find the experimental probability for each outcome.

This answer has not been graded yet.

(d) Find the theoretical probability for each outcome. (Enter your probabilities as fractions.)

(e) Compare the experimental and theoretical probabilities. Are your results the same? Explain.

2. You are given a bag with 8 green marbles, 4 blue marbles, 14 yellow marbles, and 12 red marbles. Find the theoretical probability of each random event. (Enter your probabilities as fractions.)

(a) Drawing a green marble


(b) Drawing a red marble


(c) Drawing a marble that is not yellow

1. Amy tosses 12 biased coins. Each coin comes up heads with probability 0.2. What is the probability that fewer than 3 of the coins come up heads?

Answer: 0.5583

2. Amy shoots 27000 arrows at a target. Each arrow hits the target (independently) with probability 0.2. What is the probability that at most 2 of the first 15 arrows hit the target?

Answer: 0.398

3. Amy tosses 19 biased coins. Each coin comes up heads with probability 0.1. What is the probability that more than 1 of the coins come up heads?

Answer: 0.5797

4. Amy shoots 49000 arrows at a target. Each arrow hits the target (independently) with probability 0.2. What is the probability that fewer than 3 of the first 12 arrows hit the target?

Answer: 0.5583

5. Amy rolls 16 8-sided dice. What is the probability that fewer than 1 of the rolls are 1s?

Answer: 0.1181

A building contractor is preparing a bid on a new construction project. Two other contractors will be submitting bids for the same project. Based on past bidding practices, bids from the other contractors can be described by the following probability distributions:

If required, round your answers to three decimal places.

1. If the building contractor submits a bid of $650,000, what is the probability that the building contractor will obtain the bid? Use an Excel worksheet to simulate 1,000 trials of the contract bidding process.
   
   The probability of winning the bid of $650,000 =
2. 
   
3. The building contractor is also considering bids of 675,000 and $685,000. If the building contractor would like to bid such that the probability of winning the bid is about 0.6, what bid would you recommend? Repeat the simulation process with bids of $675,000 and $685,000 to justify your recommendation.
   
   The probability of winning the bid of $675,000 =  
   The probability of winning the bid of $685,000 =  

For a certain​ candy,

55​%

of the pieces are​ yellow,

1515​%

are​ red,

55​%

are​ blue,

2020​%

are​ green, and the rest are brown.

​a) If you pick a piece at​ random, what is the probability that it is​ brown? it is yellow or​ blue? it is not​ green? it is​ striped?

​b) Assume you have an infinite supply of these candy pieces from which to draw. If you pick three pieces in a​ row, what is the probability that they are all​ brown? the third one is the first one that is​ red? none are​ yellow? at least one is​ green?

​a) The probability that it is brown is

nothing.

​(Round to three decimal places as​ needed.)The probability that it is yellow or blue is

nothing.

​(Round to three decimal places as​ needed.)The probability that it is not green is

nothing.

​(Round to three decimal places as​ needed.)The probability that it is striped is

nothing.

​(Round to three decimal places as​ needed.)​b) The probability of picking three brown candies is

nothing.

​(Round to three decimal places as​ needed.)

The probability of the third one being the first red one is

nothing.

​(Round to three decimal places as​ needed.)

The probability that none are yellow is

nothing.

​(Round to three decimal places as​ needed.)

The probability of at least one green candy is

nothing.

​(Round to three decimal places as​ needed.)

The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.04 liter. Suppose you select a random sample of 25 bottles.

a. What is the probability that the sample mean will be between 1.99 and 2.0 liters ?

b. What is the probability that the sample mean will be below 1.98 liters ?

c. What is the probability that the sample mean will be greater than 2.01 liters?

d. The probability is 99 % that the sample mean amount of soft drink will be at least how much?

e. The probability is 99 % that the sample mean amount of soft drink will be between which two values (symmetrically distributed around the mean)?

a. The probability is ____ (Round to three decimal places as needed.)

b.The probability is ____. (Round to three decimal places as needed.)

c. The probability is____. (Round to three decimal places as needed.)

d. There is a 99 % probability that the sample mean amount of soft drink will be at least ____liter(s). (Round to three decimal places as needed.)

e. There is a 99 % probability that the sample mean amount of soft drink will be between ____liter(s) and nothing liter(s). (Round to three decimal places as needed. Use ascending order.)

A product is being assembled and packaged on three production lines (line A, line B, and line C). Each day, the quality control team selects a production line and inspects a batch chosen at random from the output of the selected production line. Line A is selected with probability .5, line B is selected with probability .2, and line C is selected with probability .3. The probability that no defects will be found in a batch selected from line A is 0.98, and the corresponding probabilities for production lines B and C are 0.96 and 0.94, respectively.

a) What is the probability that the quality control team inspects a batch from line A and finds no defects?

b) What is the probability that the quality control team inspects a batch from line B and finds no defects?

c) What is the probability that the quality control team inspects a batch from line C and finds no defects?

d) What is the probability that no defects are found in any given day?

e) Given that no defects were found in a given day, what is the probability the inspected batch came from line A?

f) Given that no defects were found in a given day, what is the probability the inspected batch came from line B?

g) Given that no defects were found in a given day, what is the probability the inspected batch came from line C?