### 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.

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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.

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

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.

19. A retail establishment accepts either the American Express or VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both. What percentage of its customers carry a credit card that the establishment will accept?

This problems is from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much. 18. 45% of the students at U of M do neither have a tattoo nor a body piercing (other than an ear piercing). 40% of the students at U of M ha

The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much.

The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much.

Suppose E,F are subsets of the sample space of an experiment with random outcomes of an experiment. We often call E,F events. Define what it means for E,F to be independent. (Question also included in attachment)

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.

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?

Suppose a pair of dice are flipped. Find the probability for each of the following events: 1. The sum of the dots is even. 2. The sum of the dots is at least 5.

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*

Problem 8. Suppose that an insurance company classifies people into low, average and high risk persons. Their records indicate that the probabilities of being involved in an accident over a 1-year period are 0.05, 0.15 and 0.30 for low, average and high risk persons, respectively. Assume that 30% of the population is low-risk, 5

Prove Boole's inequalities.

A random number of dice is rolled. Find the probability that... (SEE ATTACHED)

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

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

Suppose X had probability density function cx^2 for 0 < x < 1, 0 otherwise. Find (a) the constant c, and the (b) mean, and (c) variance fo X.

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

Suppose the number of electrons X counted by an optical communication system is a poisson random variable .(see the wholeproblem in the attachment)

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

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

It was suggested that the number of particles in a randomly selected interval might follow a Poisson distribution. Assuming a Poisson distribution to be an appropriate model for the data, use two methods to find an approximate 95% confidence interval for the mean of this distribution. See attachment for full question includin

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

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.

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

Please see attachment.

In the game of bridge, a player is dealt a hand of 13 playing cards from a standard 52 card deck. A hand is said to have a void in a suit if it contains no cards in that suit. Determine the number of distinct hands containing at least one void. What is the probability of being dealt a hand with at least one void? Your answer sho

4.19) The lifetimes of two car batteries (Brand A and B) are independent exponential random variables with means 12 hours and 10 hours, respectively. What is the probability that Brand B battery outlasts Brand A battery?

3.21) Suppose a machine has three independent components with Exp(.1) lifetimes. Compute the expected lifetime of the machine if it needs all three components to function properly. (This is a multivariate random variable problem)