• 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-r where e = 2.71828… r! = np1.) 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 hospitalizedb)50% have been hospitalized

Normal Distribution-also known as the Gaussian Distribution-mathematical equation developed by De Moivre given by

- 1 (x-)2

2

P(x) = 1 e where = standard deviation

= mean = 3.14159… e = 2.71828… x = random variable

Properties of the Normal Curve1. 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% 34.13% 34.13% 13.59% 2.15%

Z -3 -2 -1 0 1 2 3

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 - = X – X

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 = 5c. 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 and a standard deviation ,

1.The distribution of sample means X is approximately normal regardless of whether or not the population distribution is normal.