please help me in the following questions and answer in detail.

An example that is given to show that a statement is not true is called a counterexample. For instance, suppose someone makes the statement "All colors are red." A counterexample to that statement would be to show someone the color blue or some other color. If a statement is always true, there are no counterexamples. The statement "All even numbers are divisible by 2" is always true. It is not possible to give an example of an even number that is not divisible by 2. In mathematics, statements that are always true are called theorerns, and mathematicians are always searching for theorerns. Sornetimes a conjecture by a mathematician appears to be a theorem. That is, the statement appears to be always true, but later on someone finds a counterexample. One example of this occurred when the French mathematician Pierre de Fermat (1601-1665) conjectured that 2(2n) + 1 is always a prime number for any natural number IT. For instance, when 17 = 3, we have and 257 is a prime number. However, in 1732 Leonhard Euler (1707-1783) showed that when n = 5, 2(25) + 4,294,967,297, and that 4,294,967,297 = 641 ? 6,700,417?without a calculator! Because 4,294,967,297 is the product of two numbers (other than itself and 1), it is not a prime number. This counterexample showed that Fermat's conjecture is not a theorem. For Exercise, answer true if the statement is always true. If there is an instance in which the statement is false, give a counterexample. 1The product of two positive numbers is always larger than either of the two nurnbers.

An example that is given to show that a statement is not true is called a counterexample. For instance, suppose someone makes the statement "All colors are red." A counterexample to that statement would be to show someone the color blue or some other color. If a statement is always true, there are no counterexamples. The statement "All even numbers are divisible by 2" is always true. It is not possible to give an example of an even number that is not divisible by 2. In mathematics, statements that are always true are called theorems, and mathematicians are always searching for theorems. Sometimes a conjecture by a mathematician appears to be a theorem. That is, the statement appears to be always true, but later on someone finds a counterexample. One example of this occurred when the French mathematician Pierre de Fermat (1601-1665) conjectured that 2(2n) + 1 is always a prime number for any natural number IT. For instance, when 17 = 3, we have 2(23) + 1 = 28 + 1 = 257, and 257 is a prime number. However, in 1732 Leonhard Euler (1707-1783) showed that when n = 5, 2(25) + 1 = 4,294,967,297, and that 4,294,967,297 = 641 . 6,700,417-without a calculator! Because 4,294,967,297 is the product of two numbers (other than itself and 1), it is not a prime number. This counterexample showed that Fermat's conjecture is not a theorem. For Exercise, answer true if the statement is always true. If there is an instance in which the statement is false, give a counterexample. The product of two positive numbers is always larger than either of the two numbers.2. A forensic chemist used the hypergeometric function in Excel, calculated the probability of drawing four playing cards from a shuffled deck and getting all aces, and reported as a percentage as a "1 in X" odds. Which result is correct? a. 1 in 270,725 b. 1 in 272,705 c. 1 in 275,720 d. 1 in 277,025Consider a modification to the stop and wait protocol. Sender sends sequence number i, the actual packet and the ERG, where] = CI for the m transmission of a packet 1 for the second time transmission, 2 for the third time transmission, and so on. The receiver sends an ACE [if it receives a packet correctly} or a NAP-i {if it receives with errors}. a} {8 points] if the receiver gives an ACE for all correctly.r received packets and decides to keep them. then show hv example that this creates duplicate packets. {Hint Assume that packet undergoes a time out and is received correctly,r on both attempts. What happens at the receiver in this case?) b] {8 points} If the receiver gives an ACE for all correctly.r received packets and a NAK for all incorrectly received packets, hut decides to keep only.r those with] = D. then show tiv example that this can result in a loss of packets. {Hint Assume packet I] is transmitted withj = It], received incorrectly.r in the rst attempt out the transmitter times out and resends packet D withj = '1. This time, assume that it is received correctlv. New show that packet [I will he lost}. c} {3 points} If the receiver gives an ACE or NAK only,r for packets with j = i]. then show that this will create a deadlock. (Hint: Assume packet [I is transmitted withj = t} and is received incorrectly. Then transmitter sees the NAB: from the receiver and retransmits packet D with] = '1. This time, what will the receiver do? Will it send an ACK? What will the transmitter do it it does not receive ACE or NAK within the time out? Now complete the scenario to show the deadlock). d] {1 point] Now argue that this protocol will not worlr..l Suppose that you go to the doctor for a regular checkup and even though you have no symptoms, the doctor requests that you be tested for a rare disease, a disease that only 1 in 10,000 people in your age group contract. The doctor informs you that the test is 99% accurate, meaning that, if you have the disease, it comes back positive 99% of the time and incorrectly produces a negative result, 1% of the time. After a week, the results come back. The doctor informs you that you have tested positive. (a) Use Bayes' theorem to calculate the probability that you have the disease. (b) Suppose that after your positive test, the doctor asks you to retake the test. Again it comes back positive. What is the probability that have the disease now? (Hint: use the answer of part (a) as the new hypothesis) (c) Suppose that instead of being symptom free, you have symptoms that are consistent with the disease. You are told by the doctor that people in your age group with symptoms will have a 1 in 12 chance of contracting the rare disease. If the test is positive, what is the probability that you have the disease