In a standard deck of 52 cards。

(a) What are the total number of five card hands?
(b) What is the probability of having exactly two Aces?
(c) What is the probability of getting a hand containing five Two’s?
(d) What is the probability of a Three given that the first four cards are not a Three?

Alex has 3 tests to take this week. The probability that Alex passes the math test is 0.63 , the probability Alex passes the History test is 0.80, and the probability Alex passes the chemistry test is 0.88

1. (4 pts) create a complete tree diagram for the three tests- let T be passed tests
2. (4 pts) create a probability distribution for the number of tests passed
3. what is the probability that Alex will fail all three tests
4. what is the probability that Alex will pass at least 2 tests

Part 2

BeerBatch Class

/**
* A class to model an item (or set of items) in an
* auction: a batch.
*/
public class BeerBatch
{
    // A unique identifying number.
    private final int number;
    // A description of the batch.
    private String description;
    // The current highest offer for this batch.
    private Offer highestOffer;

    /**
     * Construct a BeerBatch, setting its number and description.
     * @param number The batch number.
     * @param description A description of this batch.
     */
    public BeerBatch(int number, String description)
    {
        this.number = number;
        this.description = description;
        this.highestOffer = null;
    }

    /**
     * Attempt an offer for this batch. A successful offer
     * must have a value higher than any existing offer.
     * @param offer A new offer.
     * @return true if successful, false otherwise
     */
    public boolean bidFor(Offer offer)
    {
        if(highestOffer == null) {
            // There is no previous bid.
            highestOffer = offer;
            return true;
        }
        else if(offer.getAmount() > highestOffer.getAmount()) {
            // The bid is better than the previous one.
            highestOffer = offer;
            return true;
        }
        else {
            // The bid is not better.
            return false;
        }
    }
  
    /**
     * @return A string representation of this batch's details.
     */
    public String batchDetail()
    {
        return "TO DO";
    }

    /**
     * @return The batch's number.
     */
    public int getNumber()
    {
        return number;
    }

    /**
     * @return The batch's description.
     */
    public String getDescription()
    {
        return description;
    }

    /**
     * @return The highest offer for this lot.
     *         This could be null if there is
     *         no current bid.
     */
    public Offer getHighestOffer()
    {
        return highestOffer;
    }
}
Offer Class

/**
* A class that models an offer.
* It contains a reference to the Person bidding and the amount of the offer.
*/
public class Offer
{
    // The person making the bid.
    private final Bidder bidder;
    // The amount of the offer.
    private final int amount;

    /**
     * Create an offer.
     * @param bidder Who is bidding for the batch.
     * @param x The amount of the offer.
     */
    public Offer(int x, Bidder b)
    {
        this.bidder = b;
        this.amount = x;
    }

    /**
     * @return The bidder.
     */
    public Bidder getBidder()
    {
        return bidder;
    }

    /**
     * @return The amount of the offer.
     */
    public int getAmount()
    {
        return amount;
    }
}

The number of students taking the SAT has risen to an all-time high of more than 1.5 million The number of times the SAT was taken and the number of students are as follows.

a. Let x be a random variable indicating the number of times a student takes the SAT. Show the probability distribution for this random variable. Round your answers to four decimal places.

b. What is the probability that a student takes the SAT more than one time? Round your answer to four decimal places.

c. What is the probability that a student takes the SAT three or more times? Round your answer to four decimal places.

d. What is the expected value of the number of times the SAT is taken? Round your interim calculations and final answer to four decimal places.

e. What is the variance and standard deviation for the number of times the SAT is taken? Round your interim calculations and final answer to four decimal places.

The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. The distribution takes only whole-number values, so it is certainly not normal.

A) Let x-bar be the mean number of accidents at the intersection during a year (52 weeks). What is the approximate probability that x-bar is less than 2?

B) what is the approximate probability that there are fewer than 100 accidents at the intersection in a year? Hint: re-state the events in terms of x-bar

The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. The distribution takes only whole-number values, so it is certainly not normal.

A) Let x-bar be the mean number of accidents at the intersection during a year (52 weeks). What is the approximate probability that x-bar is less than 2?

B) what is the approximate probability that there are fewer than 100 accidents at the intersection in a year? Hint: re-state the events in terms of x-bar

A manager at a company that manufactures phones has noticed that the number of faulty phones in a production run of cell phones is usually small and that. the quality of one days run seems to have no bearing on the next day.
1. if the mean number of faulty cell phones is 3.9 per day, what is the probability that no faulty phones will be produced tomorrow?

2. if the mean number of faulty cell phones is 3.9 per day, what is the probability that 3 or more faulty phones were produced in today's run?

The Post Office has established a record in a major Midwestern city for delivering 90 percent of its local mail the next working day. If you mail eight local letters:

a) What is the probability that all of them will be delivered the next day?

b) What is the average number you expect to be delivered the next day?

c) Calculate the standard deviation of the number delivered when 8 local letters are mailed.

d) What is the probability that the number delivered will be within 2 standard deviations of the mean?

A company manufactures Printed Circuit Boards (PCBs) expects to have 6 defective units each day. Let Y be a random variable that counts the number of defective units produced each day.

A. Which discrete random variable distribution would best model this scenario?

B. What is the probability that the number of defective units observed in a dayexceeds the mean number by more than one standard deviation?

C. What is the probability that, on two randomly selected days, no defective units are observed?

Smith has been injured during an operation. With probability .5 the doctor is guilty of
malpractice and with probability .5 he is innocent. The doctor knows whether or not he is
actually guilty. Smith threatens to sue the doctor. Before the case goes to trial, the doctor can
settle out of court. If the doctor settles, he pays Smith 1,000, Smith pays 1/3 of this 1,000 to her
lawyer, and the doctor does not pay any legal fees. If the doctor does not settle then Smith can
either drop the suit or let the case go to trial. If Smith drops the suit Smith and the doctor each
receive a payoff of 0. If the case goes to trial, the doctor's guilt or innocence will be discovered.
If Smith wins the trial, then (i) she will win 1,000 but will have to give 1/3 of this 1,000 to her
lawyer, and (ii) the doctor will lose the 1,000 he pays Smith and an additional 100 in legal fees.
If Smith loses the trial then she must pay the doctor's legal fees of 100.

(a) Show that this game has all of the basic elements of a signaling game.

(b) Show that this game does not have a separating equilibrium.

(c) Show that this game does not have a pooling equilibrium.

(d) Find the mixed strategy perfect Bayes-Nash equilibrium in this game.