2. Two players participate in a contest where the winner gets R and the loser gets P-R, 0 ≤ R ≤ P. The two players have identical convex cost of effort functions C(e). The chance player i wins is e_i/(e₁+e₂). Show that, at equilibrium, effort increases in R.

What are your views regarding Intrinsic vs. Extrinsic motivation? Are the children who are most VULNERABLE receiving awards or attending pizza parties for good grades or appropriate behavior? Sort out your feelings and share them with the class. (Note: Someone always wins, but someone always loses in the world of "Incentives and Rewards).

A tortoise can run with a speed of 0.13 m/s, and a hare can run 20 times as fast. In a race, they both start at the same time, but the hare stops to rest for 5.0 minutes. The tortoise wins by a shell (20 cm). (a) How long does the race take? s (b) What is the length of the race?.

6.

A friend of yours thinks that he has devised a purely mathematical way of beating the standard European roulette

wheel, which has

37

pockets. His plan is to come to the roulette table with

100

units with which to bet. On each

spin, he places a one-unit, single-number bet. (Recall that winning such a bet returns

36

units to the winner.) He will

make exactly

100

bets, no matter how much he wins or loses. He claims that this strategy results in a greater than

50

%

chance that he will come out ahead at the end of

100

spins, so using this strategy repeatedly will make him a winner.

a)

Does he have a greater than

50

% chance of coming away with more money than he started with?

b)

If the answer to the previous question is yes, then does that make his strategy a winning one, meaning that it will

lead to a long-term increase in bankroll?

2)

3)

6. An officer of a micro-lending company in a developing country has made eight loans of $100 each. Let X_i be a random variable representing the real rate of return on loan i, i = 1, ..., 8. The real rate of return on loan i is a random variable because a borrower sometimes delays or skips payments, and because inflation during the life of the loan can vary. Assume X_i is approximately normally distributed with mean 0.10, and standard deviation 0.02.  Use 4 decimals in your calculations, then round to the requested number of decimals when you answer.

a. Calculate the probability that loan i will have a real rate of return from 0.09 to 0.11. This probability is  .  TWO DECIMALS.

b.  Continued. Find the symmetric interval (a, b) such that the probability is 0.99 that X_i will fall in this interval. Find the interval that is symmetric around the mean of the distribution of X_i. The value of “a” is  . The value of “b” is  .  THREE DECIMALS.

c. Calculate the probability that the mean of the real rates of return on the loan officer’s eight loans will be from 0.09 to 0.11. This probability is  .  TWO DECIMALS.

d.  Continued. Find the symmetric interval (a, b) such that the probability is 0.99 that the mean real rate of return will fall in this interval. Find the interval that is symmetric around the mean of the distribution of "Xbar". The value of “a” is  . The value of “b” is  .  THREE DECIMALS.

e. How many loans would the entrepreneur have to make in order for the probability to be at least 0.99 that the mean real rate of return will be from 0.09 to 0.11? Call this number “n”. The value of n is  .  INTEGER (NO DECIMALS).

Write a python program for the following question.

Complete the symptom similarity function, which measures the similarity between the symptoms of two patients. See below for an explanation of how the similarity is computed.

def symptom_similarity(symptoms_A: Tuple[Set], symptoms_B: Tuple[Set]) -> int:

'''Returns the similarity between symptoms_A and symptoms_B.
symptoms_A and symptoms_B are tuples of a set of symptoms present and a set of symptoms absent.
The similarity measure is computed by the following equations:
present_present + absent_absent - present_absent - absent_present
where:

present_present is the number of symptoms present in both patients

absent_absent is the number of symptoms absent in both patients

present_absent is the number of symptoms present in patientA and absent in patientB

absent_present is the number of symptoms absent in patientA and present in patientB.

>>>symptom_similarity(yang, maria)
1

#yang = ({"coughing", "runny_nose", "sneezing"},{"headache","fever"})
#maria = ({"coughing", "fever", "sore_throat", "sneezing"},{"muscle_pain"}
# As you can see the common part in these two tuple is "coughing' so similarity is 1.
''''

Question 2: similarity to patients (18 points)
Complete the similarity to patients function, which uses the function symptom similarity to select
the patients (from the entire population of patients in our database) with the highest similarity between
the their symptoms and the symptoms of one patient. See below for an explanation of exactly what is
expected:

def similarity_to_patients(my_symptoms : Tuple[Set], all_patients: Dict[str, Tuple[Set]]) ->
Set[int]:
"""
returns a set of the patient IDs with the highest similarity between my_symptoms
(which is a tuple of symptoms present and absent) and the symptoms of
all the patients stored in the dictionary all_patients.
>>> similarity_to_patients(yang, all_patients_symptoms)
{45437, 73454}
#PATIENTS ARE ASSIGNED ID and those ID stores their symptoms. So we need to compare "yang's" symptoms to other patients symptoms in the database.
"""

I will drop a like for even attempting to give a fruitful solution.

Arrange these biological molecules (nucleic acid, cellulose, unsaturated fatty acid, saturated fatty acid) in order from highest to lowest energy density? Take note of which chemical bonds have the highest potential energy.  Can you explain which bonds have high or low potential energy and why? Also, covalent bonds between which of the atoms will have the highest potential energy?

1. A poll is given, showing 20% are in favor of a new building project.
If 10 people are chosen at random, what is the probability that exactly 4 of them favor the new building project?

2. According to Masterfoods, the company that manufactures M&M’s, 12% of peanut M&M’s are brown, 15% are yellow, 12% are red, 23% are blue, 23% are orange and 15% are green. You randomly select six peanut M&M’s from an extra-large bag of the candies. (Round all probabilities below to four decimal places; i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.)
a.) Compute the probability that exactly two of the six M&M’s are yellow.
b.) Compute the probability that two or three of the six M&M’s are yellow.
c.) Compute the probability that at most two of the six M&M’s are yellow.
d.) Compute the probability that at least two of the six M&M’s are yellow.

3. A small regional carrier accepted 11 reservations for a particular flight with 10 seats. 7 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 40% chance, independently of each other.
a.) Find the probability that overbooking occurs.
b.) Find the probability that the flight has empty seats

4. The Smith family was one of the first to come to the U.S. They had 7 children. Assuming that the probability of a child being a girl is .5, find the probability that the Smith family had:a.) at least 3 girls?
b.) at most 3 girls?

5. A company prices its tornado insurance using the following assumptions:
• In any calendar year, there can be at most one tornado.
• In any calendar year, the probability of a tornado is 0.14.
• The number of tornadoes in any calendar year is independent of the number of tornados in any other calendar year.
Using the company's assumptions, calculate the probability that there are fewer than 3 tornadoes in a 10-year period.

14. The table below ~hows the probabilities of all realizations
for each of the three dice. Notice that the dice are all
"loaded" meaning that they are unbalanced in some way
leading to an unequal probability of any number coming
up.

1. If die 3 is rolled, what is the probability of rolling a
two?
b. If one of these three dice is selected at random and
roll ed, what is the probability of rolling a two?
c. If one of these three dice is selected at random and
rolled, what number is the least likely to appear?
d. If one of these three dice is selected at random and a
two is rolled, what is the probability that die 3 was
rolled?
e. If one of these three dice is selected at random and a
four is rolled, what die is the most likely to have been
chosen?
f. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is 7?
g. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is less than S?
h. If die 1 and 2 are selected and rolled, and one adds
their realizations, what is the sum is that is more
likely to appear?
i. If die 2 and 3 are selected and rolled, and one adds
their realizations, what is the probability that this
sum is less than 11?
j. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is 2?
k. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is 5?
I. If die 1 and 3 are selected and rolled, what is the
probability that the minimum of the two realizations
is less than 4?
m. If die 2 and 3 are selected and rolled, what is the
probability that the maximum of the two realizations
is greater than 6?