In the New York State "Numbers" game, three digits are drawn, with replacement, each evening on television. The player chooses, in advance of course, three digits and may pick from several types of bets:

a. "Straight": To win the player's digits must match those drawn in the order they were drawn.

b. "Six-Way Box": The player chooses three different digits and wins if they match those drawn in any order.

c. "Three-Way Box": The player announces a digit once and a different digit twice. To win, these three digits must match the three digits drawn in any order.

Find the probability of winning each of these bets.

The seniors at Weseltown High School are voting on where to go for their senior trip. They are deciding on Angel Falls(A), Bend Canyon(B), Cedar Lake(C) or Danger Gap(D). The results of the preferences are:

DABC = 120

ACBD = 100

BCAD = 90

CBDA = 80

CBAD = 45

a. Who is the Condorcet candidate?

b. Is there a majority winner? If not, is there a plurality winner? Does this violate the Condorcet criterion?

c. Who wins the Borda count? Does this violate the Condorcet criterion?

d. Who wins using the Hare method? Does this violate the Condorcet criterion?

e. Who wins using the pairwise comparison method? Does this violate the Condorcet criterion?

The developers of a new online game have determined from preliminary testing that the scores of players on the first level of the game can be modelled satisfactorily by a Normal distribution with a mean of 185 points and a standard deviation of 28 points. They would like to vary the difficulty of the second level in this game, depending on the player’s score in the first level.

(a) The developers have decided to provide different versions of the second level for each of the following groups:

(i) those whose score on the first level is in the lowest 25% of scores

ii) those whose score on the first level is in the middle 50% of scores

(iii) those whose score on the first level is in the highest 25% of scores. Use the information given above to determine the cut-off scores for these groups. (You may round each of your answers to the nearest whole number.)

(b) In the second level of the game, the developers have also decided to give players an opportunity to qualify for a bonus round. Their stated aim is that players from group (i) should have 75% chance of qualifying for the bonus round, players from group (ii) should have 55% chance of qualifying for the bonus round and that players from group (iii) should have 30% chance of qualifying for this round. Let ?, ? and ? respectively denote the events that a player’s score on the first level was in the lowest 25% of scores, the middle 50% of scores and the highest 25% of scores, and let ? denote the event that the player qualifies for the bonus round. Use event notation to express the developers’ aim as a set of conditional probabilities.

(c) Based on the developers’ stated aim, find the total probability that a randomly chosen player will qualify for the bonus round.

(d) Given that a player has qualified for the bonus round, what is the probability that the player’s score on the first level was in the middle 50% of scores for that level?

(e) Given that a player has not qualified for the bonus round, what is the probability that the player’s score on the first level was in the lowest 25% of scores for that level?

The developers of a new online game have determined from preliminary testing that the scores of players on the first level of the game can be modelled satisfactorily by a Normal distribution with a mean of 185 points and a standard deviation of 28 points. They would like to vary the difficulty of the second level in this game, depending on the player’s score in the first level. (a) The developers have decided to provide different versions of the second level for each of the following groups: (i) those whose score on the first level is in the lowest 25% of scores ii) those whose score on the first level is in the middle 50% of scores (iii) those whose score on the first level is in the highest 25% of scores. Use the information given above to determine the cut-off scores for these groups. (You may round each of your answers to the nearest whole number.) (b) In the second level of the game, the developers have also decided to give players an opportunity to qualify for a bonus round. Their stated aim is that players from group (i) should have 75% chance of qualifying for the bonus round, players from group (ii) should have 55% chance of qualifying for the bonus round and that players from group (iii) should have 30% chance of qualifying for this round. Let ?, ? and ? respectively denote the events that a player’s score on the first level was in the lowest 25% of scores, the middle 50% of scores and the highest 25% of scores, and let ? denote the event that the player qualifies for the bonus round. Use event notation to express the developers’ aim as a set of conditional probabilities. (c) Based on the developers’ stated aim, find the total probability that a randomly chosen player will qualify for the bonus round. (d) Given that a player has qualified for the bonus round, what is the probability that the player’s score on the first level was in the middle 50% of scores for that level? (e) Given that a player has not qualified for the bonus round, what is the probability that the player’s score on the first level was in the lowest 25% of scores for that level?

- Part 2 – 4 Desk Checking Exercises – these are 3 programs (pseudocode) that are working and 1 program (Python). Explainthe intent of the pseudocode / program. If you use test data, note the test data.  .  You are NOT trying to find mistakes.

What does this do? Desk Checking #2:  Explain the intent of this pseudocode.  Be as specific as possible.
List the data you use as the example data.

start

         guess number between 1 and 100

         while guess is not correct

                           if guess is too high

                                             guess a number lower than the previous guess

                           else

                                             guess a number higher than the previous guess

                           endif

         endwhile                 

         player wins

stop

The mean cost of domestic airfares in the United States rose to an all-time high of $385 per ticket (Bureau of Transportation Statistics website, November 2, 2012 ). Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $110.

a. What is the probability that a domestic airfare is $550 or more (to 4 decimals)?

b. What is the probability that a domestic airfare is $250 or less (to 4 decimals)?

c. What if the probability that a domestic airfare is between $300 and $500 (to 4 decimals)?

d. What is the cost for the 3% highest domestic airfares? (rounded to nearest dollar)

The mean cost of domestic airfares in the United States rose to an all-time high of $385 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $120. Use Table 1 in Appendix B. a. What is the probability that a domestic airfare is $560 or more (to 4 decimals)? b. What is the probability that a domestic airfare is $260 or less (to 4 decimals)? c. What if the probability that a domestic airfare is between $310 and $510 (to 4 decimals)? d. What is the cost for the 2% highest domestic airfares? (rounded to nearest dollar) $ or more/less.