A rare form of malignant tumor occurs in 11 children in a​ million, so its probability is 0.000011. Four cases of this tumor occurred in a certain​ town, which had

15915 children.

a. Assuming that this tumor occurs as​ usual, find the mean number of cases in groups of

15915 children.

b. Using the unrounded mean from part

​(a​),

find the probability that the number of tumor cases in a group of

15915

children is 0 or 1.

c. What is the probability of more than one​ case?

d. Does the cluster of four cases appear to be attributable to random​ chance? Why or why​ not?

a. The mean number of cases is

?

​(Type an integer or decimal rounded to three decimal places as​ needed.)

b. The probability that the number of cases is exactly 0 or 1 is

?

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

c. The probability of more than one case is

?

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

d. Let a probability of 0.05 or less be​ "very small," and let a probability of 0.95 or more be​ "very large". Does the cluster of four cases appear to be attributable to random​ chance? Why or why​ not?

A.

​No, because the probability of more than one case is very small.

Your answer is correct.

B.

​Yes, because the probability of more than one case is very small.

C.

​Yes, because the probability of more than one case is very large.

D.

​No, because the probability of more than one case is very large.

According to an​ airline, flights on a certain route are on time 85​% of the time. Suppose 25 flights are randomly selected and the number of​ on-time flights is recorded.

​(a) Explain why this is a binomial experiment.

​(b) Find and interpret the probability that exactly17 flights are on time.

​(c) Find and interpret the probability that fewer than17 flights are on time.

​(d) Find and interpret the probability that at least 17 flights are on time.

​(e) Find and interpret the probability that between 15 and 17 flights, inclusive, are on time.

​(a) Identify the statements that explain why this is a binomial experiment. Select all that apply.

A.There are three mutually exclusive possibly​ outcomes, arriving​ on-time, arriving​ early, and arriving late.

B.The trials are independent.

C.The experiment is performed until a desired number of successes is reached.

D.The probability of success is the same for each trial of the experiment.

E.There are two mutually exclusive​ outcomes, success or failure.

F.The experiment is performed a fixed number of times.

G.Each trial depends on the previous trial.

​(b) The probability that exactly 17 flights are on time is

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

Interpret the probability.

In 100 trials of this​ experiment, it is expected about _ to result in exactly 17 flights being on time.

​(Round to the nearest whole number as​ needed.)

​(c) The probability that fewer than 17 flights are on time is _

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

Interpret the probability.

In 100 trials of this​ experiment, it is expected about _ to result in fewer than 17 flights being on time.

​(Round to the nearest whole number as​ needed.)

​(d) The probability that at least 17 flights are on time is _

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

Interpret the probability.

In 100 trials of this​ experiment, it is expected about _ to result in at least 17 flights being on time.

​(Round to the nearest whole number as​ needed.)

​(e) The probability that between 15 and 17 flights, inclusive, are on time is _

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

Interpret the probability.

In 100 trials of this​ experiment, it is expected about _ to result in between 15 and 17 flights, inclusive, being on time.

Mr. Jackson could think about his experiment a little differently. How many times should he expect to roll a 4 if he rolls his number cube 30 times? This is called the mean, or expected value. You can find it if you know the number of trials and the probability of success for an individual trial. Use the formula μx = np to find the expected value.

Part A: Find the expected value for the number of times you would roll a 4 in 30 rolls of a number cube.

a. How many trials, n, are there?

b. What is p, the probability of success for each trial?

c. What is the expected value of rolling a 4 in 30 rolls of a number cube?

Part B: You can also find the standard deviation for the binomial probability distribution of a specific outcome in a binomial experiment. Use the formula to find the standard deviation. You've already identified n and p in Part A. Show your work, and round your answer to two decimal places.

Part C: You've seen that a cumulative binomial probability is the probability of getting a number of successes within a given range. It's easiest to use your calculator's binomcdf command to find these. Find each of the probabilities described below. Show your work, and round your answer to three
decimal places.

a. What is the probability that Mr. Jackson will roll a 4 at most 5 times out of his 30 rolls of a
number cube?

b. What is the probability that Mr. Jackson will roll a 4 at least 5 times out of his 30 rolls of a number cube?

c. What is the probability that Mr. Jackson will roll a 4 between 3 and 7 times, inclusive, out of his 30 rolls of a number cube? Use this formula: P(a ≤ x ≤ b) = binomcdf(n,p,b) – binomcdf(n,p,a – 1).

1. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is (a) replaced, (b) not replaced.
2. Find the probability of a 4 turning up at least once in two tosses of a fair die.

3. One bag contains 4 white balls and 2 black balls; another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that (a) both are white, (b) both are black,(c) one is white and one is black.

4. Box I contains 3 red and 2 blue marbles while Box II contains 2 red and 8 blue marbles. A fair coin is tossed. If the coin turns up heads, a marble is chosen from Box I; if it turns up tails, a marble is chosen from Box II. Find the probability that a red marble is chosen.

5. A committee of 3 members is to be formed consisting of one representative each from labor, management, and the public. If there are 3 possible representatives from labor,2 from management, and 4 from the public, determine how many different committees can be formed

6. In how many ways can 5 differently colored marbles be arranged in a row?

7. In how many ways can 10 people be seated on a bench if only 4 seats are available?

8. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

9. How many 4-digit numbers can be formed with the 10 digits 0,1,2,3,. . . ,9 if (a) repetitions are allowed, (b) repetitions are not allowed, (c) the last digit must be zero and repetitions are not allowed?

10. Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if (a) the books in each particular subject must all stand together, (b) only the mathematics books must stand together?

11. Five red marbles, two white marbles, and three blue marbles are arranged in a row. If all the marbles of the same color are not distinguishable from each other, how many different arrangements are possible?

12. In how many ways can 7 people be seated at a round table if (a) they can sit anywhere,(b) 2 particular people must not sit next to each other?

13. In how many ways can 10 objects be split into two groups containing 4 and 6 objects, respectively?

14. In how many ways can a committee of 5 people be chosen out of 9 people?

15. Out of 5 mathematicians and 7 physicists, a committee consisting of 2 mathematicians and 3 physicists is to be formed. In how many ways can this be done if (a) any mathematician and any physicist can be included, (b) one particular physicist must be on the committee, (c) two particular mathematicians cannot be on the committee?

16. How many different salads can be made from lettuce, escarole, endive, watercress, and chicory?

17. From 7 consonants and 5 vowels,how many words can be formed consisting of 4 different consonants and 3 different vowels? The words need not have meaning.

18. In the game of poker5 cards are drawn from a pack of 52 well-shuffled cards. Find the probability that (a) 4 are aces, (b) 4 are aces and 1 is a king, (c) 3 are tens and 2 are jacks, (d) a nine, ten, jack, queen, king are obtained in any order, (e) 3 are of any one suit and 2 are of another, (f) at least 1 ace is obtained.

19. Determine the probability of three 6s in 5 tosses of a fair die.

20. A shelf has 6 mathematics books and 4 physics books. Find the probability that 3 particular mathematics books will be together.

21. A and B play 12 games of chess of which 6 are won by A,4 are won by B,and 2 end in a draw. They agree to play a tournament consisting of 3 games. Find the probability that (a) A wins all 3 games, (b) 2 games end in a draw, (c) A and B win alternately, (d) B wins at least 1 game.

22. A and B play a game in which they alternately toss a pair of dice. The one who is first to get a total of 7 wins the game. Find the probability that (a) the one who tosses first will win the game, (b) the one who tosses second will win the game.

23. A machine produces a total of 12,000 bolts a day, which are on the average 3% defective. Find the probability that out of 600 bolts chosen at random, 12 will be defective.

24. The probabilities that a husband and wife will be alive 20 years from now are given by 0.8 and 0.9, respectively. Find the probability that in 20 years (a) both, (b) neither, (c) at least one, will be alive.

ok, I'll update this to 3 to 4 questions. Thanks!

. (1) A man has a 4-sided (tetrahedral) die, with four faces, 1, 2, 3, and 4. He rolls it 12 times. (a) Given that for each roll of the die, the probability of rolling a 4 is 1/4 and the probability of not doing so is 3/4, compute the following probabilities P(x = x0),where x is the number of times that the man rolls a 4 among the 12 trials. (b) What is the average value of x? What is the standard deviation? (c) Which possible value of x has the highest likelihood of occurring? (2) IQ scores in America, by definition, are normally distributed, with 100 points being the mean IQ and the standard deviation being 10 points. (a) Suppose a person has an IQ of 115. What percentage of the population is expected to have a higher IQ than this person? (b) Now suppose a person has an IQ of 75. What percentage of the population is expected to have a lower IQ than this person? (c) An individual’s IQ was claimed to have jumped from 83 to 212 after a procedure. (An IQ of 212 is not possible, given that most tests only can assign scores of at most 170.) What percentage of the population, were the claim not false, would be between the first score and the second?

Player I and player II each have two pennies. Each player holds 0, 1, or 2 pennies in his left hand and the remainder of the pennies (2, 1, or 0 respectively) in his right hand. Each player reveals both hands simultaneously. If the number of coins in one of player I's hands is greater than the number of coins in the respective hand of player II, player I wins the difference in pennies; otherwise, no money is exchanged.

(a). Find the von Neumann value and the optimal strategy for each player in the game. If any player has infinitely many optimal strategies, find all optimal strategies.

(b). If player I owed $100 to player II, approximately how many rounds of the game would have to be played, on the average, to cancel the debt?

Suppose you are deciding how to invest $2,000. There are two
options: shares in an airline company and shares in a petroleum company. There is a 0.25
probability that oil prices will go up, a 0.25 probability that prices will go down, and a 0.50
percent probability they will remain the same. The value of each $100 invested after prices
adjust is as follows:

a. Calculate the expected value and standard deviation of the value of the total investment if
you invest the entire $2,000 in the airline company.
b. Calculate the expected value and standard deviation of the total investment if you invest
the entire $2,000 in the oil company.
c. Calculate the expected value and standard deviation of the total investment if you invest
$1,000 in the airline and $1,000 in the oil company.
d. Which investment (‘a’, ‘b’, or ‘c’) has the highest risk? Which has the lowest risk?

A company has a total of 30 employees.

17 employeers graduated from Berkeley.

Out of 30 employees, 18 are native Californians. Furthermore, the company has 12 mechanical
engineers. Still, only 1 employee can claim the highest honor of being a native Californian, Berkely graduate mechanical engineer.

(a)If an employee is selected at random from this company, what is the probability
that this employee is a Berkely graduate?

(b)A survey reveals that, out of 12 mechanical engineering working at this company,
half are native Californians. How many Californians are not mechanical engineers?

(c)Suppose that a native Californian is selected at random. The probability that this
employee is an Berkeley graduate is 1/2. If a Berkeley graduate is chosen at random, what is the probability that this employee is a native Californian?

(d)When the Berkeley graduates come together for a basketball game, there are only 3
mechanical engineers in the room. If an employee is selected at random from this company,
what is the probability that this employee is an mechanical engineer but neither a Berkeley graduate
nor a native Californian?

Please show work

QUESTION 1

Studd Enterprises sells big-screen televisions. A concern of management is the number of televisions sold each day. A recent study revealed the number of days that a given number of televisions were sold.

                        # of TV units sold      # of days

                                       0                             2

1.                         4

2.                       10

3.                       12

4.                         8

5.                         4

Answer the questions below. For each part, show your calculations and/or explain briefly how you arrived at your answer, as appropriate or needed.

Required:

1. Convert the frequency distribution above into a probability distribution (or relative frequency distribution) showing the proportion of days (rather than the number of days) that the number of televisions sold was 0, 1, 2, 3, 4, and 5 respectively. and Compute the mean of this general discrete probability distribution. ( 6 marks)
2. Compute the standard deviation of this general discrete probability distribution. ( 5 marks)
3. What is the probability that exactly 4 televisions will be sold on any given day? and What is the probability that 2 or more televisions will be sold on any given day? ( 2 mark)
4. What is the probability that less than 2 televisions will be sold on any given day? and What is the probability that between 1 and 4 televisions inclusive will be sold on any given day? (2 mark)