probability problem

Jim and andrew are throwing darts at a target, and Wayne's probability of hitting the bullseye is ? and Yifu's probability of hitting it is ?

(independently of Wayne). A round of the game is for Yifu to throw and then Wayne to throw. The game is to keep throwing until both of them hit the bulleye on the same round and then stop.

(A) If ?

= the number of rounds until the game stops, what is the distribution of ?

?

(B) What is the probability that the game stops on the ??ℎ

round?

(C) What is the probability that Yifu first hits the target in the 4th round, but Wayne has not yet hit the target by the 4th round?

(D) Suppose after 10 rounds (and no round where both have hit the target) they decide to change the rules and continue to play until at least one of them hits the target. How many more rounds would they expect to play on average?

You must express your answers in terms of the parameters ?

and ? (and ? for (B)).

An airline knows that in the long run only 90% of passengers who book a seat show up for their flight. On a particular flight with 193 seats there are 225 reservations. Let X denote the number of passengers that show up. (a) (4 pts) Identify the probability distribution of X by name and the parameters needed. (b) (6 pts) Assuming passengers make independent decisions, what is the exact probability that the flight will be over-booked? Do not simplify your answer! (c) (8 pts) Approximate the probability in the previous part using an appropriate distribution. Justify your approximation, and find the numerical value of that.

Frost Bank finds that 24 customers arrive at the single drive-through per hour. The teller is able to complete 32 transactions per hour. Assume M/M/1 operating characteristics.

1. What is the average time a customer spends waiting?

2. What is the probability that the system is idle?

3. What is the probability that there are more than 4 customers in line?

4. What is the average number of customers in line?

5. How long does the average customer spend in the system?

6. What is the utilization factor?

7. What is the probability that a customer will have to wait?

Five males with an​ X-linked genetic disorder have one child each. The random variable x is the number of children among the five who inherit the​ X-linked genetic disorder. Determine whether a probability distribution is given. If a probability distribution is​ given, find its (a) mean and (b) standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied.

X | P(x)

0 | 0.033

1|0.156

2 | 0.311

3 | 0.311

4 | 0.156

5 | 0.033

Find the mean of the random variable x.

Find the Standard deviation of the random variable x.

Traffic flow is traditionally modeled as a Poisson Distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of 9 vehicles per minute.

What is the random variable X described above? Write the distribution of X using the standard notations.                                          

What should be the variance and the standard deviation of the number of vehicles through the intersection within a minute?                              

What is the probability that exactly 12 vehicles will pass the intersection within a minute?   

What is the probability that 10 or less than 10 vehicles will pass the intersection within a minute?
  

What is the probability that more than 10 vehicles will pass the intersection within a minute?
  

An office has a stock of identical printed forms which are used independently. On any working day, at most one of the forms is used, and the probability that one form is used is 1/3 . There are 250 working days in the year.

(i) Using a suitable approximation, calculate the number of forms that must be in stock at the beginning of the year if there is to be a 95% probability that they will not all be used before the end of the year. [5 marks]

(ii) If one form in one hundred is unusable due to faulty printing and these faults occur at random, calculate the probability that in a batch of 250 forms there will be not more than one which is unusable.

1. A round in the game Yahtzee begins by rolling five fair dice. Find the probability of rolling a:

a. one pair (ex 33421 but not 33441), two pair (ex 33441 but not 33444), and three of a kind (ex 24252 but not 24242)

2. Consider a 10x10 matrix that consists of all zeros. ten elements of the matrix are selected at random and their value is changed from a zero to a one. find the probability that the ones fall in a line (row-wise, column-wise, or diagonally)

3. Five fair dice are rolled simultaneously. Find the probability that the total number of spots on the up faces totals 28 or more

6. Of all the televisions that are sent to a certain workshop for repair, 80% are no longer under warranty

a. What is the expected number of televisions under warranty for a group of 20 televisions? (5 pts)

b. Among those 20 televisions, what is the probability that at least 75% are not covered under the warranty? (8pts)

c. Suppose there are currently 12 televisions in the workshop of which 5 are under warranty. 8 televisions are taken at random to be worked

i. What is the probability that at least 4 of those that are under warranty will be chosen? (5pts)

ii. What is the probability that all televisions of the same type will be chosen? (7pts)