Hi,Please help me.

Rolling a Die. If we repeatedly roll a balanced die, then, in
the long run, it will come up “4” about one-sixth of the time. But
what is the probability that such a die will come up “4” exactly
once in six rolls?

Gestation Periods. The probability is 0.314 that the gestation
period of a woman will exceed 9 months. In six human
births, what is the probability that the number in which the gestation
period exceeds 9 months is
a. exactly three?
b. exactly five?
c. at least five?
d. between three and five, inclusive?

Recidivism. In the Scientific American article “Reducing
Crime: Rehabilitation is Making a Comeback,” R. Doyle examined
rehabilitation of felons. One aspect of the article discussed
recidivism of juvenile prisoners between 14 and 17 years old, indicating
that 82% of those released in 1994 were rearrested within
3 years. Suppose that, today, six newly released juvenile prisoners
between 14 and 17 years old are selected at random.
a. Assuming that the recidivism rate is the same today as it was
in 1994, determine the probability distribution for the number,
Y , who are rearrested within 3 years.
b. Determine and interpret the mean of Y .
c. If, in fact, exactly two of the six newly released juvenile prisoners
are rearrested within 3 years, would you be inclined to
conclude that the recidivism rate today has decreased from the
82% rate in 1994? Explain your reasoning. Hint: First consider
the probability P(Y = 2).
d. If, in fact, exactly four of the six newly released juvenile prisoners
are rearrested within 3 years, would you be inclined to
conclude that the recidivism rate today has decreased from the
82% rate in 1994? Explain your reasoning.

Lotto. A previous Arizona state lottery called Lotto is
played as follows: The player selects six numbers from the numbers
1–42 and buys a ticket for $1. There are six winning numbers,
which are selected at random from the numbers 1–42. To
win a prize, a Lotto ticket must contain three or more of the winning
numbers. A probability distribution for the number of winning
numbers for a single ticket is shown in the following table.
Number of
winning numbers Probability
0 0.3713060
1 0.4311941
2 0.1684352
3 0.0272219
4 0.0018014
5 0.0000412
6 0.0000002
a. If you buy one Lotto ticket, determine the probability that you
win a prize. Round your answer to three decimal places.

b. If you buy one Lotto ticket per week for a year, determine the
probability that you win a prize at least once in the 52 tries.

Sickle Cell Anemia. Sickle cell anemia is an inherited
blood disease that occurs primarily in blacks. In the United
States, about 15 of every 10,000 black children have sickle cell
anemia. The red blood cells of an affected person are abnormal;
the result is severe chronic anemia (inability to carry the required
amount of oxygen), which causes headaches, shortness of breath,
jaundice, increased risk of pneumococcal pneumonia and gallstones,
and other severe problems. Sickle cell anemia occurs in
children who inherit an abnormal type of hemoglobin, called
hemoglobin S, from both parents. If hemoglobin S is inherited
from only one parent, the person is said to have sickle cell trait
and is generally free from symptoms. There is a 50% chance that
a person who has sickle cell trait will pass hemoglobin S to an
offspring.
a. Obtain the probability that a child of two people who have
sickle cell trait will have sickle cell anemia.
b. If two people who have sickle cell trait have five children, determine
the probability that at least one of the children will
have sickle cell anemia.
c. If two people who have sickle cell trait have five children, find
the probability distribution of the number of those children
who will have sickle cell anemia.
d. Construct a probability histogram for the probability distribution
in part (c).
e. If two people who have sickle cell trait have five children, how
many can they expect will have sickle cell anemia?

Tire Mileage. A sales representative for a tire manufacturer
claims that the company’s steel-belted radials last at least
35,000 miles. A tire dealer decides to check that claim by testing
eight of the tires. If 75% or more of the eight tires he tests
last at least 35,000 miles, he will purchase tires from the sales
representative. If, in fact, 90% of the steel-belted radials produced
by the manufacturer last at least 35,000 miles, what is the
probability that the tire dealer will purchase tires from the sales
representative?

Restaurant Reservations. From past experience, the
owner of a restaurant knows that, on average, 4% of the parties
that make reservations never show. How many reservations can
the owner accept and still be at least 80% sure that all parties that
make a reservation will show?
5.74 Sampling and the Binomial Distribution. Refer to the
discussion on the binomial approximation to the hypergeometric
distribution that begins on page 234.
a. If sampling is with replacement, explain why the trials are independent
and the success probability remains the same from
trial to trial—always the proportion of the population that has
the specified attribute.
b. If sampling is without replacement, explain why the trials are
not independent and the success probability varies from trial
to trial.

Sampling and the Binomial Distribution. Following is
a gender frequency distribution for students in Professor Weiss’s
introductory statistics class.
Gender Frequency
Male 17
Female 23
Two students are selected at random. Find the probability that
both students are male if the selection is done
a. with replacement.
b. without replacement.
c. Compare the answers obtained in parts (a) and (b).
Suppose that Professor Weiss’s class had 10 times the students,
but in the same proportions, that is, 170 males and 230 females.
d. Repeat parts (a)–(c), using this hypothetical distribution of
students.
e. In which case is there less difference between sampling without
and with replacement? Explain why this is so.

The Poisson Distribution. Another important discrete
probability distribution is the Poisson distribution, named in
honor of the French mathematician and physicist Simeon Poisson
(1781–1840). This probability distribution is often used to
model the frequency with which a specified event occurs during
a particular period of time. The Poisson probability formula is
P(X = x) = e-? ?x
x!
,
where X is the number of times the event occurs and ? is a parameter
equal to the mean of X. The number e is the base of natural
logarithms and is approximately equal to 2.7183.
To illustrate, consider the following problem: Desert Samaritan
Hospital, located in Mesa, Arizona, keeps records of emergency
room traffic. Those records reveal that the number of patients
who arrive between 6:00 P.M. and 7:00 P.M. has a Poisson
distribution with parameter ? = 6.9. Determine the probability
that, on a given day, the number of patients who arrive at the
emergency room between 6:00 P.M. and 7:00 P.M. will be
a. exactly 4.
b. at most 2.
c. between 4 and 10, inclusive

The Challenger Disaster. In a letter to the editor that appeared
in the February 23, 1987, issue of U.S. News and World
Report, a reader discussed the issue of space-shuttle safety. Each
“criticality 1” item must have a 99.99% reliability, by NASA
standards, which means that the probability of failure for a “criticality
1” item is only 0.0001. Mission 25, the mission in which
the Challenger exploded on takeoff, had 748 “criticality 1” items.
Use the Poisson approximation to the binomial distribution to determine
the approximate probability that
a. none of the “criticality 1” items would fail.
b. at least one “criticality 1” item would fail.

Fragile X Syndrome. The second-leading genetic cause
of mental retardation is Fragile X Syndrome, named for the
fragile appearance of the tip of the X chromosome in affected
individuals. One in 1500 males are affected worldwide, with no
ethnic bias.
a. In a sample of 10,000 males, how many would you expect to
have Fragile X Syndrome?
b. For a sample of 10,000 males, use the Poisson approximation
to the binomial distribution to determine the probability that
more than 7 of the males have Fragile X Syndrome; that at
most 10 of the males have Fragile X Syndrome.

Holes in One.According to the experts, the odds against a PGA golfer making a hole in one are 3708 to 1; that is, the probability is 1
3709 .Use the Poisson approximation to the binomial distribution
to determine the probability that at least 4 of the 155 golfers
playing the second round would get a hole in one on the
sixth hole.


Rolling a Die. If we repeatedly roll a balanced die, then, in
the long run, it will come up “4” about one-sixth of the time. But
what is the probability that such a die will come up “4” exactly
once in six rolls?