Borb b about to play a two-game che:s:s match with an opponent, and wants to find the strategy that maximizes his winning chances. Each game ends with either a win by one of the players, or a draw. If the score is tied at the end of the two games, the match goes into sudden-death mode, and the players continue to play until the first time one of them wins a game (and the match). Boris has two playing styles. timid and bold, and he can choose one of the two at will in each game. no matter what style he chose in previous games. With timid play. he draws with probability Pd > 0, and he loses with probability 1 -Pd. With bold play. he wins with probability pw, and he loses with probability 1 -pw' Boris will always play bold during sudden death, but may switch style between games 1 and 2. (a) Find the probability that Boris wins the match for each of the following strategies: (i) Play bold in both games 1 and 2. (ii) Play timid in both games 1 and 2. (iii) Play timid whenever he is ahead in the score. and play bold otherwise. (b) Assume that pw < 1/2, so Boris is the worse player, regardless of the playing style he adopts. Show that with the strategy in (iii) above. and depending on the values of pw and Pd. Boris may have a better than a 50-50 chance to win the match. How do you explain this advantage?

5.

The 2024 Olympics have just concluded! The local news has reported that the Arstotzkan Olympic team has

won exactly three medals, but has not said which ones.

Since we are unsure what the specific medal results are, we can create a sample space to represent the possible numbers of bronze, silver, and gold medals that are won. We can summarize the outcomes by abbreviating these medals by their first letter, so 2B0S1G represents the outcome where the team won two bronze medals, no silver medals, and one gold medal. Our sample space would be:

S = {3B0S0G, 2B0S1G, 2B1S0G, 1B0S2G, 1B1S1G, 1B2S0G, 0B0S3G, 0B1S2G, 0B2S1G, 0B3S0G}
(a) If we assume that this is an equiprobable sample space, determine the probability that at least one gold medal

is won by the Arstotzkan Olympic team.
(b) Let X be a random variable which counts the number of silver medals won by the Arstotzkan team, write

out the events, X ≤ 1, X > 2, and X = 4, as collections of sample points in S.
(c) Instead of assuming the sample space is equally likely, we assume the following:

1. all the outcomes where the Arstotzkan team wins at least one gold medal are equally likely

2. all the outcomes where the Arstotzkan team wins no gold medals are equally likely

3. outcomes where the Arstotzkan team wins at least one gold medal are twice as likely as outcomes where the Arstotzkan team wins no gold medals

Using this assumption, create a p.d.f for X, the random variable from (b).

Objective

To explore the details of Ethernet frames. Ethernet is a popular link layer protocol that is covered in §4.3 of your text; modern computers connect to Ethernet switches (§4.3.4) rather than use classic Ethernet (§4.3.2). Review section §4.3 before doing this exercise.

Requirements

Wireshark: This lab uses the Wireshark software application to capture and examine a packet trace. A packet trace is a record of traffic at a location on the network, as if a snapshot was taken of all the bits that passed across a particular wire. The packet trace records a timestamp for each packet, along with the bits that make up the packet, from the lower-layer headers to the higher-layer contents. Wireshark shows the sequence of packets and the meaning of the bits when interpreted as protocol headers and data. It color-codes packets by their type, and has various ways to filter and analyze packets to let you investigate the behavior of network protocols.

ping: This exercise uses ping to send and receive messages. ping is a standard command-line utility for checking that another computer is responsive. It is widely used for network troubleshooting and comes pre-installed on Window, Linux, and Mac. While ping has various options, simply issuing the command “ping www.sdsu.edu” will cause your computer to send a small number of ICMP ping requests to the remote computer www.sdsu.edu, which should elicit an ICMP ping response.

Recall that there are two types of Ethernet frames, IEEE 802.3 and DIX Ethernet. DIX is common and what we considered above, while IEEE 802.3 is rare. If you are rather lucky, you may see some IEEE 802.3 frames in the trace you have captured. To search for IEEE 802.3 packets, enter a display filter (above the top panel of the Wireshark window) of “llc” (that was lowercase “LLC”) because the IEEE 802.3 format has the LLC protocol on top of it. LLC is also present on top of IEEE 802.11 wireless, but it is not present on DIX Ethernet.

Have a look at the details of an IEEE 802.3 frame, including the LLC header. Observe that the Type field is now a Length field. The frame may be short enough that there is also padding of zeros identified as a Trailer or Padding.

1. How long are the combined IEEE 802.3 and LLC headers compared to the DIX Ethernet headers? Note that the Trailer/Padding and Checksum may be shown as part of the header, but they come at the end of the frame.
2. How does the receiving computer know whether the frame is DIX Ethernet or IEEE 802.3? Hint: you may need to both use Wireshark to look at packet examples and read your text near where the Ethernet formats are described.
3. If IEEE 802.3 has no Type field, then how is the next higher layer determined? Use Wireshark to look for the demultiplexing key.

Thomas’ manager has a deal with his employees that he will buy lunch on Friday for the highest weekly performer on his team if that employee wins at least two out of three coin flips. If the highest performer loses at least two out of three coin flips, then that employee has to buy lunch for the manager. To perform the three flips, three quarters are flipped simultaneously, so everyone gets to see three outcomes (heads or tails) each time. In 40 weeks, the manager has won the flips 27 times and Thomas suspects that the quarters being used are unfair as a result. He never trusted his manager, so he’s been keeping track since the first contest, and he has tracked the following results over the 40 weeks:

If the quarters were fair, then that would mean that n_heads comes from a binomial distribution with p = 0.5 (and number of trials = 3 – note that number of trials is a fixed parameter for the binomial, not an estimated parameter, so it doesn’t affect the degrees of freedom for the chi-squared test).

Using a chi-squared test for goodness of fit, can we say with 95% confidence that the quarters are unfair (i.e., that these results do not come from a binomial distribution with p=0.5)?

Can we say with 90% confidence that the quarters are unfair?

Flaca has brought a bathroom scale with her in an elevator and is standing on it. Just before the elevator arrives at the top floor, as the car is slowing down, she notices that according to the scale, her weight appears to be off by 3% from its normal value W.

What is the magnitude of the acceleration of the elevator?

A 67-kg woman in an elevator is accelerating downward at a rate of 1.8 m/s2. What is the magnitude of the force exerted by the elevator floor on the woman's feet?

A 51-kg woman in an elevator is accelerating downward at a rate of 1.9 m/s2. What is the magnitude of the gravitational force acting upon the woman?

A 82-kg man stands on a spring scale in an elevator. Starting from rest, the elevator ascends, attaining its maximum speed of 1.2 m/s in 0.74 s. The elevator travels with this constant speed for 5.0 s, undergoes a uniform negative acceleration for 1.4 s, and then comes to rest.

(a) What does the spring scale register before the elevator starts to move?
  N

(b) What does the spring scale register during the first 0.74 s of the elevator's ascent?
  N

(c) What does the spring scale register while the elevator is traveling at constant speed?
  N

(d) What does the spring scale register during the elevator's negative acceleration?
  N

An elevator rail is assumed to meet specifications if its diameter is between 0.98 and 1.01 inches. Each year a company produces 100, 000 elevator rails. For a cost of $10/σ2 per year the company can rent a machine that produces elevator rails whose diameters have a standard deviation of σ. (The idea is that the company must pay more for a smaller variance.) Each such machine will produce rails having a mean diameter of one inch. Any rail that does not meet specifications must be reworked at a cost of $12. Assume that the diameter of an elevator rail follows a normal distribution. Round your answers to three decimal places, if necessary.

a. What standard deviation (within 0.001 inch) minimizes the annual cost of producing elevator rails? You do not need to try standard deviations in excess of 0.02 inch.

b. For your answer in part a, one elevator rail in 1000 will be at least how many inches in diameter?

You are designing a high-speed elevator for a new skyscraper. The elevator will have a mass limit of 2400 kg (including passengers). For passenger comfort, you choose the maximum ascent speed to be 18.0 m/s, the maximum descent speed to be 10.0 m/s, and the maximum acceleration magnitude to be 2.70 m/s2. Ignore friction. A). What is the maximum upward force that the supporting cables exert on the elevator car? B). What is the minimum upward force that the supporting cables exert on the elevator car? C). What is the minimum time it will take the elevator to ascend from the lobby to the observation deck, a vertical displacement of 640 m? D).What is the maximum value of a 60.0-kg passenger’s apparent weight during the ascent? E). What is the minimum value of a 60.0-kg passenger’s apparent weight during the ascent? F). What is the minimum time it will take the elevator to descend to the lobby from the observation deck, a vertical displacement of 640 m?