Transcript Example

PROBABILITY AND
STATISTICS
WEEK 5
Onur Doğan
The Binomial Probability Distribution
There are many experiments that conform either exactly or
approximately to the following list of requirements:
1. The experiment consists of a sequence of n smaller experiments
called trials, where n is fixed in advance of the experiment.
2. Each trial can result in one of the same two possible outcomes
(dichotomous trials), which we denote by success (S) and failure (F).
3. The trials are independent, so that the outcome on any particular trial
does not influence the outcome on any other trial.
4. The probability of success is constant from trial to trial; we denote
this probability by p.
An experiment for which Conditions 1–4 are satisfied is called a
binomial experiment.
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Bernoulli trials
A Bernoulli refers to a trial that has only two possible outcomes.
(1) Flipping a coin: S = {head, tail)
(2) Truth of an answer: S = {right, wrong)
(3) Status of a machine: S = {working, broken)
(4) Quality of a product: S = {good, defective)
(5) Accomplishment of a task: S = {success, failure)
A binomial experiment consists of a series of n independent
Bernoulli trials with a constant probability of success (p) in
each trial.
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Example
• A seller’s success ?
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The Mean and Variance of X
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Example
• Suppose that a machine produce defective item
with probability 0,1.
a) Suppose that we selected 5 items, find the probability
of 1 item defective.
b) If the amount of daily production is 100, then what's
the expected defective item amount?
c)What’s the variance of defective items of samples
around the expected defective items.
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Example
The probability of making a doctor's successful
surgery is %80.
If that doctor make 3 surgery in one month, find
the all probability for all possible results.
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Example
 In a certain automobile dealership, 20% of all customers purchase
an extended warranty with their new car. For 7 customers selected
at random:
1) Find the probability that exactly 2 will purchase an
extended warranty
2) Find the probability at most 6 will purchase an
extended warranty
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Example
 Example: Find the mean and standard deviation of the binomial
distribution when n = 18 and p = 0.75. Define the probability function.
Solutions:
1) n = 18,
p = 0.75,
q = 1 - 0.75 = 0.25
m = np = (18)(0.75) = 13.5
s =
npq =
( 18 )(0. 75 )(0. 25 ) =
3 .375  1.8371
2) The probability function is:
 18 
P ( x ) =  ÷ (0. 75 ) x (0. 25 ) 18- x for
 x 
x = 0, 1, 2, . . . , 18
The Hypergeometric Distribution
The assumptions leading to the hypergeometric distribution are as follows:
1. The population or set to be sampled consists of N individuals,
objects, or elements (a finite population).
2. Each individual can be characterized as a success (S) or a failure
(F), and there are M successes in the population.
3. A sample of n individuals is selected without replacement in such a
way that each subset of size n is equally likely to be chosen.
The random variable of interest is X the number of S’s in the sample. The
probability distribution of X depends on the parameters n, M, and N, so we
wish to obtain P(X x) h(x; n, M, N).
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Example
Suppose that a box contains five red balls and
ten blue balls. If seven balls are selected at
random without replacement, what is the
probability that three red balls will be obtained?
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The Hypergeometric Distribution
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Example
• Suppose that in production line for every 20
products, 4 of them enter reprocessing.
a) If we selected 2 products, find the probability
of one of them enter reprocessing?
b) If we selected 10 products, how many of them
should have expected enter reprocessing?
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Note:
The hypergeometric distribution is related to the
binomial distribution.
Whereas the binomial distribution is the
approximate probability model for sampling
without replacement from a finite dichotomous
(S–F)
population,
the
hypergeometric
distribution is the exact probability model for the
number of S’s in the sample.
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The Negative Binomial Distribution
The negative binomial rv and distribution are based on an
experiment satisfying the following conditions:
1. The experiment consists of a sequence of independent trials.
2. Each trial can result in either a success (S) or a failure (F).
3. The probability of success is constant from trial to trial,
so P(S on trial i)=p for i =1, 2, 3 . . . .
4. The experiment continues (trials are performed) until a total of
r successes have been observed, where r is a specified positive
integer.
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The Negative Binomial Distribution
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Example
A pediatrician wishes to recruit 5 couples, each
of whom is expecting their first child, to
participate in a new natural childbirth regimen.
Let p=P(a randomly selected couple agrees to
participate).
If p=0.2, what is the probability that 15 couples
must be asked before 5 are found who agree to
participate?
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The Negative Binomial Distribution
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The Geometric Distributions
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Example
• In a production line 200 of 1000 items were found
to be defective.
a)What’s the probability of first defective item is the
4th item tested.
b)How many items should have been tested till first
defective item found?
c)What’s the probability of the first defective item is
not the first tested one?
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The Multinomial Distributions
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Example
Suppose that there are 3 different brand; A,B and C.
And we have probabilities to be purchased;
P(A)=0,40
P(B)=0,10
P(C)=0,50
Suppose that there are 10 customers, what’s the
probability of 2 of them buy A, 5 of them buy B
and 3 of them buy C.
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The Poisson Probability Distribution
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Example
Suppose that, in İzmir the number of power blackout
has the Poisson distribution with mean 2, for one year.
• Find the probability of there will be no power blackout in next
year?
• Find the probability of there will be 2 power blackout in next 6
months?
• Find the probability of there will be 2 or more blackout in next
year?
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Example
The number of requests for assistance received by a towing service
is a Poisson process with rate =4 per hour.
a. Compute the probability that exactly ten requests are received
during a particular 2-hour period.
b. If the operators of the towing service take a 30-min break for
lunch, what is the probability that they do not miss any calls for
assistance?
c. How many calls would you expect during their break?
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