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# Probability

### Probability : Games Won

1. Suppose that two teams play a series of games that ends when one of the teams has one i number of games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when (a) i = 2 and when (b) i = 3. Show also in both cases that this number is maximized

### Probability : Bernoulli Trials and Exponential Distributions

3.3. Assume that within a given service game at tennis, successive points form Bernoulli trials with p = P(Server wins) > ½. Tennis rules say that the service game ends as soon as either player has won at least four points, and is at least two points ahead of the other. Find the chances the server wins the game 4-0, 4-1, and 4-

### Probability : Probability Generating Function and Beta and Gamma Function

3.2 A majority verdict of 10-2 or better may be permitted in a jury trial. Assuming each juror has probability 0.9 of reaching a Guilty verdict, and decides independently, what is the probability the jury decides to convict? (Partial solution provided in attachment) 3.14 For both the Gamma and Beta distribution, find the va

### Probability : General Probability, Systems and Dice

2.17 Show that, if (a) a fair die is thrown times independently, it is more likely than not that at least one six appears; (b) a pair of fair dice are thrown 24 times independently, it is more likely than not that a double six does not appear. (This pair of calculation has an honoured place in the history of the developmen

### Probability : Bayes' Theorem and General Probability

For a different medical application of Bayes' theorem, suppose one person in 1000 suffers an adverse reaction to a drug, and a simple test for this reaction is on offer. The test is said to be 95% reliable, meaning that if the person would suffer a reaction, a positive result comes up 95% of the time, and if they would not have

### Probability

The probability that a family with 6 children has exactly four boys is: a. 1/3 b. 1/64 c. 15/64 d. 3/8 e. none of the above

### Probability : What is the best strategy?

18. A true-false question is to be posed to a husband and wife team on a quiz show. Both the husband and the wife will, independently, give the correct answer with probability p. which of the following is a better strategy for the couple? (a) Choose one of them and let that person answer the question; or (b) have them both con

### Probability

6. If there are 12 strangers in a room, what is the probability that no two of them celebrate their birthday in the same month?

### Probability: jailer's reasoning

40. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already know

### Proof : Probability - Independent Events

Prove or give a counterexample. If E1 and E2 are independent, then they are conditionally independent given F. Please see the attached file for the fully formatted problem.

### Posterior Probability

An absent-minded nurse is to give Mr. Brown a pill each day. The probability that the nurse forgets to administer the pill is 2/3. If he receives the pill, the probability that Mr. Brown will die is 1/3. If he does not get his pill, the probability that he will die is 3/4. Mr. Brown dies. What is the probability that the nurse f

### Finding Probability with Given Restrictions

12. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G: (a) only E occurs; (b) both E and G but not F occur; (c) at least one of the events occurs; (d) at least two of the events occur; (e) all three occur; (f) none of the events occurs.

### Find the Winning Strategy for a Coin Flip Game

Your opponent specifies 3 successives results of tosses of a coin, e.g. HHT. You then specify another such result, e.g. THT. The winner is the person whose sequence appears first when a fair coin is tossed successively and independently. Find the strategy which will allow you, the second player, to win at least 2/3 of the time.

### Child Gender Probability

Assuming boy and girl children are equally likely and births are independent, if parents have two children, what is the probability that at least one is a boy? On the condition that at least one is a boy, what is the probability that one of the children is a girl?

Please use words to describe the solution process. (I've attached a possibly useful review of probability). Suppose that P is finitely additive on an alegbra {field} A. Show that P is countably additive on A <---> whenever {An} is a sequence in ... *see attachment*

### Probability

34. Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked as an exponential random variable with parameter 1/20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 2

### Binomial distribution sample sizes

Your lab is working to produce a particular chemical reaction. The conventional probability for producing this reaction successfully is p = ½. You have a new technique that you believe will produce this reaction successfully at least 2/3 of the time. You plan to test your method with a sequence of 36 trials. You decide to r

### Probability- lifetime of light bulbs

A certain factory has three machines A,B and C that produces one type of light bulbs.Past experience has shown that the lifetime of a light bulb by machine A can be modeled as an exponential random variable with an average of 20 days, whereas the lifetime of a bulb produced by a machine B can modeled as a normal random variable

### Probability- The game continues until soneone gets heads. What is the probability of each of them winning?

Al, Bob and Carlos are playing a silly game. Al flips a coin. If he gets heads, the game ends and he wins. If not, Bob flips the coin. If he gets heads, the game ends and he wins. If not, Carlos flips the coin. If he gets heads, the game ends and he wins. If not, the coin is returned to Al and the entire process begins a

### Geometry : Probability that Three Points on a Circle will form a Right-Triangle

If n points are equally spaced on the circumference of a circle, what is the probability that three points chosen at random will form a right triangle? I know that for us to have a right triangle, the two points should form the diameter of the circle. What I have done is that I divided the problem into two sections. Section

### A sample of n independent observations...

Please see the attached file for full problem description. --- ? A sample of n independent observations are obtained of a random variable having a Poisson distribution with mean . Show that the maximum likelihood estimate of is he sample mean show that the corresponding estimator is an unbiased estimator of , and h

### Dice problem involving probability

Constuct an unique pair of dice (6 sides) so that each of the sums 2 through 12 has an equal (nonzero i.e. cannot have two dice with all zero's) probability of occurring. The dice do not have to be identical.

### Probabilities

Question 1 A restaurant can serve up to 75 meals. Experience shows that 20% of clients who have booked do not turn up. 1. The manager accepts 90 bookings. What is the probability that more than 50 clients turn up? 2. How many bookings should the manager accept in order to have a probability of more than 0.9 that he will s

### Sample derivative of double integral

Problem Note: the problem is part of a thesis I'm working on. Define: > v > 0, where and v are parameters (constant). v < a < , where a is a parameter (constant). v1 - variable F(·) - probability distribution function with support [v, ]. f(·) = F'(·) - probability density function, strictly positiv

### Probability using Bayes theorem

1.22) An oil executive has determined that the probability that this oil field contains oil is 0.6. Before starting the drilling she decides to order a seismological test to see if there is oil in the ground. Unfortunately, the test is not entirely accurate. It concludes that there is oil with probability 0.9 if there is indeed

### Probability

Let E and F be non-zero-probability events. If E and F are mutually-exclusive, can they also be independent? Explain the answer, and also prove it algebraically using the definitions of mutually-exclusive and independent events.

### Probability

An experimental test to detect a particular type of cancer indicates the presence of cancer in 90% of individuals known to have this type of cancer, and in 15% of individuals known to be cancer-free (false positive). One hundred individuals volunteer to take the test. Of the 100, 60 are known to have the cancer, and 40 are known

### Joint Probability Questions

See attachment Newspaper article frequently cite the fact that in any one year a small percentage (say 10%) of all drivers are responsible for all automobile accidents. The conclusion is often reached that if only we could single out these accident-prone drivers and either retrain them or remove them from the reads we could d

### Probability : Target Shooting

An archer has probability 0.3 of hitting a certain target. What is the probability of hitting the target exactly two times in four attempts?

### Probability : Random Selection

In a carnival game the players selects two coins from a bag containing two silver dollars and six slugs. Write down the probability distribution for the winnings and determine how much the player would have to pay so that he would break even, on the average, over many repetitions of the game.