Transcript MONTHLY STARTING SALARY (In TRL)

• Example:
Consider a recent study conducted by the personnel manager of
a major computer software company.
The study showed that 30% of employees who left the firm
within two years did so primarily because they were dissatisfied
with their salary, 20% left because they were dissatisfied with
their work assignments, 12% of the former employees indicated
dissatisfaction with both their salary and their work assignments.
• Question:
What is the probability that an employee who leaves within two
years does so because of dissatisfaction with salary,
dissatisfaction with work assignment or both?
Assigning Probabilities
 Basic Requirements for Assigning Probabilities
2. The sum of the probabilities for all experimental
outcomes must equal 1.
P(E1) + P(E2) + . . . + P(En) = 1
where:
n is the number of experimental outcomes
Multiplication Law
The multiplication law provides a way to compute the
probability of the intersection of two events.
The law is written as:
P(A B) = P(B)P(A|B)
Mutual Exclusiveness and Independence
Do not confuse the notion of mutually exclusive
events with that of independent events.
Two events with nonzero probabilities cannot be
both mutually exclusive and independent.
If one mutually exclusive event is known to occur,
the other cannot occur.; thus, the probability of the
other event occurring is reduced to zero (and they
are therefore dependent).
Two events that are not mutually exclusive, might
or might not be independent.
The Sales of Automobiles for 300 days
0
automobile sold 54 days
1
automobile sold 117 days
2 automobile sold 72 days
3 automobile sold 42 days
4 automobile sold 12 days
5 automobile sold 3 days
Total: 300 days
Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), that provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
Example-1:
An insurance company sells a 10,000 TRL 1-year term insurance
policy at an annual premium of 290 TRL. Based on many year’s
information, the probability of death during the next year for a
person of customer’s age, sex, health etc. is 0.001
Q: What is the expected gain (amount of money made by the
company) for a policy of this type?
Example: 2
The College Board website provides much information for students, parents, and
professionals with respect to the many aspects involved in Advanced Placement (AP)
courses and exams. One particular annual report provides the percent of students who
obtain each of the possible AP grades (1 through 5). The 2008 grade distribution for all
subjects was as follows:
AP Grade Percent
1
20.9
2
21.3
3
24.1
4
19.4
5
14.3
a. ) Express this distribution as a discrete probability distribution.
b. ) Find the mean and standard deviation of the AP exam scores for 2008.
Properties of the Binomial Probability Distributions
1- The experiment consists of a sequence of n identical trials
2- Two outcomes (SUCCESS and FAILURE ) are possible on
each trial
3- The probability of success, denoted by p, does not change
from trial to trial. Consequently, the probability of failure,
denoted by q and equals to 1-p , does not change from trial to
trial
4- The trials are independent.
Example :4
The Heart Association claims that only 10% of adults over 30 can pass the minimum
requirements of Fitness Test. Suppose four adults are randomly selected and each is
given the fitness test.
Use the formula for a binomial random variable to find the probability distribution of
x, where x is the number of adults who pass the fitness test. Graph the distribution.
Properties of the Binomial Probability Distributions
1- The experiment consists of a sequence of n identical trials
2- Two outcomes (SUCCESS and FAILURE ) are possible on
each trial
3- The probability of success, denoted by p, does not change
from trial to trial. Consequently, the probability of failure,
denoted by q and equals to 1-p , does not change from trial to
trial
4- The trials are independent.
Example :4 Fitness Test
The Heart Association claims that only 10% of adults over 30 can pass the minimum
requirements of Fitness Test. Suppose four adults are randomly selected and each is
given the fitness test.
Use the formula for a binomial random variable to find the probability distribution of
x, where x is the number of adults who pass the fitness test. Graph the distribution.
Example: Purchase Decision
Consider the purchase decisions of the next three customers who enter
the clothing store. On the basis of past experience, the store manager
estimates the probability that any one customer will make a purchase is
0.30
Q: What is the probability that two of the next three customers will make a
purchase?
Properties of Normal Distributions
1- The entire family of normal distribution is differentiated by its mean µ and
its standard deviation σ.
2- The highest point on the normal curve is at the mean which is also the
median and the mode of the distribution.
3- The mean of the distribution can be any numerical value: negative, zero or
positive.
4- The normal distribution is symmetric
5- The standard deviation determines how flat and wide the curve is
6- Probabilities for the random variables are given by areas under the curve.
The total area under the curve for the normal distribution is 1
7- Because the distribution is symmetric, the area under the curve to the left of
the mean is 0.50 and the area under the curve to the right of the mean is 0.50
8- The percentage of values in some commonly used intervals are;
a-) 68.3% of the values of a normal r.v are within plus or minus one st.dev. of
its mean
b-) 95.4% of the values of a normal r.v are within plus or minus two st.dev. of
its mean
c-) 99.7% of the values of a normal r.v are within plus or minus three st.dev. of
its mean
Find the area under the standard normal curve that lies
1- to the right of z = - 0.55
2- to the left of
z=
0.84
3- to the right of z =
1.69
4- to the left of
z = - 0.74
5- between z = 0.90 and z= 1.33
6- between z = -0.29 and z= 0.59
A sociologist has been studying the criminal justice system in a large city. Among other
things, she has found that over the last 5 years the length of time an arrested person
must wait between their arrest and their trial is a normally distributed variable x with
µ=210 days and σ=20 days.
Q-) What percent of these people had their trial between 160 days and 190 days after
their arrest?
Q-1 - MOBILE PHONE
Assume that the length of time, x, between charges of mobile phone is normally
distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the
probability that the mobile phone will last between 8 and 12 hours between charges.
Q-2 ALKALINITY LEVEL
The alkalinity level of water specimens collected from the River in a country has a mean
of 50 milligrams per liter and a standard deviation of 3.2 milligrams per liter. Assume
the distribution of alkalinity levels is approximately normal and find the probability that
a water specimen collected from the river has an alkalinity level
A-) exceeding 45 milligrams per liter.
B-) below 55 milligrams per liter.
C-) between 51 and 52 milligrams per liter.
The grades of 400 students in a statistics course are normally
distributed with mean µ = 65 and variance σ2=100
Q: Find the probability that a student selected randomly from this
group would score within any interval given below.
1- A grade between 60 and 65
2- A grade between 70 and 65
3- A grade between 52 and 68
4- A grade that is greater than 85
5- A grade that is less than 72
6- A grade between 70 and 78
Example : 1
A normal distribution has the mean 74.4
Find its standard deviation if 10% of the area under the curve lies to the right of
100
Example : 2
A random variable has a normal distribution with standard deviation 10 . Find its
mean if the probability is 0.8264 that it will take on a value less than 77.5
Example :3
For a certain random variable having the normal distribution, the probability is
0.33 that it will take on a value less than 245 and the probability is 0.48 that it will
take on a value greater than 260.
Find the mean and standard deviation of the random variable