• Probability
-The ratio of the number of ways the specified event
can occur to the total number of equally likely events
that can occur.
P(E) = n = number of favorable outcomes
N number of possible outcomes
Mutually Exclusive Events
- events that cannot happen simultaneously;that is,if
one event happens, the other event cannot happen.
0 < P(Ej) < 1
P(E1) + P(E2) + … + P(En) = 1
P(not E1) = 1- P(E1)
where E1,E2 ,…,En are mutually exclusive
events .
Binomial Distribution
-also known as the Bernoulli Distribution.
-is concerned with an outcome which can
happen only in 2 different ways,i.e.,either a
“success” denoted by p or a “failure” denoted by
q.
P(r ) =
n! prqn-r
r!(n-r)!
where : n = the number of trials
r = the number of successes
n-r = the number of failures
p = the probability of success
q = 1-p , the probability of failure
1)About 50% of all persons three years of age
and older wear glasses or contact lenses.For
a randomly selected group of five people,
compute the probability that
a) exactly three wear glasses or contact
lenses
b)at least one wears them
c)at most one wears them
2. If 50% of all children born in a certain
hospital are boys,what is the probability that
among 8 children born on one day there are
3 boys and 5 girls?
3. If the probability that a patient will survive
a disease is 0.90, find the probabilities that
among 4 patients having this disease 0, 1, 2,
3 or 4 will survive.
Poisson Distribution
-the limiting form of the binomial distribution
where the probability of success is very small and
the number of trials is very large.
P(r) = e-mmr
where e = 2.71828…
r!
m= np
1.) It is known that approximately 2% of the
population is hospitalized at least once during a
year.If 100 people in such a community are to be
interviewed ,what is the probability that you will
find
a)exactly three have been hospitalized
b)50% have been hospitalized
Normal Distribution
-also known as the Gaussian Distribution
-mathematical equation developed by De Moivre
given by
- 1 (x-m)2
2
s2
P(x) =
where
1
e
s 2
s = standard deviation
m = mean
= 3.14159…
e = 2.71828…
x = random variable
Properties of the Normal Curve
1. It is symmetrical about X.
2. The mean is equal to the median, which
is also equal to the mode.
3. The tails or ends are asymptotic relative to
the horizontal line.
4. The total area under the normal curve is
equal to 1 or 100%.
5. The normal curve area may be subdivided
into at least three standard scores each to the left
and to the right of the vertical axis.
6. Along the horizontal line, the distance from
one integral standard score to the next integral
standard score is measured by the standard
deviation.
Area under the Normal Curve
99.74%
95.45%
68.26%
2.15% 13.59%
Z
34.13%
-3 -2 -1
34.13% 13.59%
2.15%
0
1
2
3
m-3s
m-2s
m-1s
m
m+1s
m+2s
m+3s
X-3s
X-2s
X-1s
X
X+1s
X+2s
X+3s
z- score(Standard score)
- gives the relative position of any
observation in a normal distribution.
z = X -m = X–X
s
s
1.Find the area under the normal curve that lies
between the given values of z.
a. z = 0 and z = 2.37
d. z = -3 and z = 3
b. z = 0 and z = -1.94
e. z = 5
c. z = -1.85 and z = 1.85
2. If the heights of male youngsters are normally
distributed with a mean of 60 inches and a
standard deviation of 10, what percentage of the
boy’s heights (in inches) would we expect to be
a. between 45 and 75;
b. between 30 and 90 ;
c. less than 50;
d. 45 or more;
e. 75 or more ;
f. between 50 and 75?
Sampling Distribution of Means
Distribution of sample means- set of values of
sample means obtained from all possible samples
of the same size(n) from a given population.
Central Limit Theorem
For a randomly selected sample of size n ( n
should be at least 25, but the larger n is,the better
the approximation)with a mean m and a standard
deviation s ,
1.The distribution of sample means X is
approximately normal regardless of whether or not
the population distribution is normal.