Ken has a coin that has probability 1/5 of landing Heads.

   Mary has a coin that has probability 1/3 of landing Heads.

   They toss their coins simultaneously, repeatedly.

Let X be the number of tosses until Ken gets his first Heads.

Let Y be the number of tosses until Mary gets her first Heads. Find:

Let U = min(X,Y) and V = max(X,Y)

(d) For k = 1, 2, 3,... , find a formula for P(U = k).

(e) For k = 1, 2, 3,... , find a formula for P(V > k). HINT: Inclusion-Exclusion.

Suppose certain coins have weights that are normally distributed with a mean of

5.641 g5.641 g

and a standard deviation of

0.069 g0.069 g.

A vending machine is configured to accept those coins with weights between

5.5215.521

g and

5.7615.761

g.

a. If

300300

different coins are inserted into the vending​ machine, what is the expected number of rejected​ coins?The expected number of rejected coins is

2525.

​(Round to the nearest​ integer.)b. If

300300

different coins are inserted into the vending​ machine, what is the probability that the mean falls between the limits of

5.5215.521

g and

5.7615.761

​g?The probability is approximately

(missing data)

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

Use R to answer the following question. Copy and paste the code and answer from R into your paper.

On the average,five cars arrive at a particular car wash every hour. Let X count the number of cars that arrive from 10 AM to 11 AM. Then X ∼pois(lambda = 5). Also, μ = σ2 = 5.
 What is the probability that no car arrives during this period?
 Suppose the car wash above is in operation from 8AM to 6PM, and we let Y be the number of customers that appear in this period. Since this period covers a total of 10 hours. What is the probability that there are between 48 and 50 customers, inclusive?

Example :

In rolling a die once , suppose that the event A is getting

an even number and the event B is getting a prime number and the event C is getting the five calculate the following probabilities :

1) P(A) , P(B) , P( C )

2) ??∪?, ??∩?, ? ?∩??, ???, ? (??P(A∪B), P(A∩B), P( A∩B^c ), P(A^c ), p (B^c)  

-------------------------

Example :

A coin is tossed three times , suppose that the event A is the heads is more

Than the tails come up and the event B is first toss results in tails .

Answer the following question :

1)Find the sample space by using tree diagram

2) Calculate

P(A) , P(B) ,??∪?, ??∩?, ?( ?∩??),P(A∪B), P(A∩B), P( A∩B^c) , ?(?)?〖p(A)〗^c

----------------------------------------------

Example :

When you with draw a card from a set of 20 cards numbered from { 1 - 20 }

Answer the following question :

1) The probability of obtaining number that is divisible by 5 or accepts

Division 7 ?

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Example :

Find the probability of events in the dice experiment one ?

Answer the following question :

1) Getting a number less than 6

2) Getting a number equal 6

3) Obtain a negative number

4) Obtain a positive number

5) Getting an even number

6) Getting an odd number

At an early stage of clinical trials of a certain method of gender​ selection, 14 couples using that method gave birth to 13 boys and 1 girl. Complete parts​ (a) through​ (e) below.

a. Assuming that the method has no effect and boys and girls are equally​ likely, use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high​ (for the number of boys in 14 ​births). Based on the​ results, is the result of 13 boys significantly high​?

Significantly low values for the number of boys in 14 births are _____ and lower. Significantly high values for the number of boys in 14 births are ____ and higher. The result of 13 boys (is or is not?) significantly high. ​(Type integers or decimals rounded to one decimal place as​ needed.)

b. Find the probability of exactly 13 boys in 14 ​births, assuming that the method has no effect. _______ ​(Type an integer or decimal rounded to six decimal places as​ needed.)

c. Find the probability of 13 or more boys in 14 ​births, assuming that the method has no effect. _____ ​(Type an integer or decimal rounded to six decimal places as​ needed.)

d. Which probability is relevant for determining whether 13 boys is significantly high​: the probability from part​ (b) or part​ (c)? Based on the relevant​ probability, is the result of 13 boys significantly high​?

The probability from (Part C or Part B) is relevant for determining whether 13 boys is significantly high. Based on this​ probability, the result of 13 boys (is or is not) significantly high.

e. What do the results suggest about the effectiveness of this method of gender​ selection?

A. The results do not suggest anything about the effectiveness of this method of gender selection.

B. The results suggest that this method is not effective at transforming the distribution of genders to a binomial distribution.

C. The results suggest that this method is effective at transforming the distribution of genders to a binomial distribution. D. The results suggest that this method is not effective in increasing the likelihood that a baby is a boy.

E. The results suggest that this method is effective in increasing the likelihood that a baby is a boy.

Use the following information to answer problems 1 through 5: Han Solo guesses randomly at

six multiple choice questions on an exam. Each question has four potential answers.

Page

2

of

2

7. This scenario can be modeled as a

(a) normal experiment with 4 trials and success probability of 1/6 per trial.

(b) normal experiment with 6 trials and success probability of 1/4 per trial.

(c) binomial experiment with 6 trials and success probability of 1/5 per trial.

(d) binomial experiment with 4 trials and success probability of 1/6 per trial.

(e) binomial experiment with 6 trials and success probability of 1/4 per trial.

8. Complete the following table that represents the probability distribution of

X

= the number of

questions Han guesses correctly.

x

0

1

2

3

4

5

6

P(X = x)

(a) 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1

(b) .1780, .5339, .8306, .9624, .9954, .9996, 1

(c) 1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7

(d) .1780, .3560, .2966, .1318, .0330, .0044, .0002

(e) 1/6, 1/6, 1/6, 1/6, 1/6, 1/6

9. What is the probability he will answer at least 4 of the questions correctly?

(a) 0.0376 (b) 10/12 (c) .67% (d) 2/12 (e) 0.0046

10. What is the mean number of questions he will answer correctly?

(a) 0 (b) 1 (c) 1.5 (d) 2 (e) 4

11. What is the standard deviation of the number of questions he will answer correctly?

(a) 0.46 (b) 1 (c) 1.06 (d) 1.5 (e) 1.6

PROBLEM 5. A box contains 10 tickets labeled 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Draw four tickets and find the probability that the largest number drawn is 8 if:

(a) the draws are made with replacement.

(b) the draws are made without replacement.

PROBLEM 6. Suppose a bakery mixes up a batch of cookie dough for 1,000 cookies. If there are raisins in the dough, it's reasonable to assume raisins will independently have a .001 chance of ending up in any particular cookie (assuming that raisins don't clump together), so that the number of raisins in a cookie behaves like a binomial (n= # raisins in whole batch, p= .001 ) random variable.

(a). How many raisins should be put into a batch of dough to make the chance of a cookie containing at least one raisin very close to 99% ? Calculate this using the binomial distribution.

(b). If there is a 99% probability that a cookie contains at least one raisin, what is the probability that a cookie contains exactly one raisin? Exactly 4 raisins? Calculate these using the binomial distribution, with the n you found in part (a).

(c). Now assume that the number of raisins in a cookie is Poisson, with mean mu. What is the right value of mu that makes the probability of at least one raisin in a cookie equal to 99% ?

(d). Assuming that the number of raisins in a cookie is Poisson with the mu that you found in part (c), what is the probability that a cookie contains exactly one raisin? Exactly 4 raisins?

REMARK There's something a bit subtle going on in this cookie story. For example, if you put 2,000 raisins into the dough and make 1,000 cookies, with raisins independently equally likely to end up in any of the 1,000 cookies, then of course the sum over all 1,000 cookies of the raisins-per-cookie numbers will add up to exactly 2,000. However, the fractions of cookies with exactly k raisins will, with very high probability, be close to Poisson (mu=2) probabilities, so those fractions would behave as though the numbers of raisins in cookies were independent Poisson (mu=2) random variables. If the numbers of raisins per cookie really were independent Poisson (mu=2) random variables, then the total number of raisins would be Poisson with mean 2,000 (which is approximately normal, with mean 2,000 and standard deviation SQRT(2,000) = 44.72). It turns out that the exact joint behavior of the rasins per cookie numbers with 2,000 raisins total is the same as the joint behavior with independent Poisson (mu=2) raisins per cookie, conditional on the total number of raisins being exactly 2,000.

In airline​ applications, failure of a component can result in catastrophe. As a​ result, many airline components utilize something called triple modular redundancy. This means that a critical component has two backup components that may be utilized should the initial component fail. Suppose a certain critical airline component has a probability of failure of 0.033 and the system that utilizes the component is part of a triple modular redundancy.

​(a) What is the probability that the system does not​ fail?

​(b) Engineers decide to the probability of failure is too high for this system. Use trial and error to determine the minimum number of components that should be included in the system to result in a system that has greater than a 0.99999999 probability of not failing

You are presented with 400 coins. 250 of them are fair coins, while the remaining 150 land heads with probability 0.65.

Part a: If you select 60 of the coins at random, what is the probability that less than half of them are fair coins?

Part b: What is the probability that a randomly selected coin flipped once will land heads?

Part c: Consider the following procedure:

1. Select one of the coins randomly.

2. Flip the coin.

3. Record whether the coin lands tails.

4. Replace the coin and thoroughly mix the coins.

If this procedure is repeated 100 times, what is the probability that the number of times that the coin lands tails will be less than 40?

You are presented with 400 coins. 250 of them are fair coins, while the remaining 150 land tails with probability 0.60.
Part a: If you select 60 of the coins at random, what is the probability that less than half of them are fair coins?
Part b: What is the probability that a randomly selected coin flipped once will land tails?
Part c: Consider the following procedure:
1. Select one of the coins randomly.
2. Flip the coin.
3. Record whether the coin lands heads.
4. Replace the coin and throroughly mix the coins.
If this procedure is repeated 100 times, what is the probability that the number of times that the coin lands heads will be less than 40?