Probabilities and Statistics presentation.

1. PROBABILITY & PROBABILITY DISTRIBUTION
STATISTICAL METHODS FOR HEALTH ADMINISTRATION (QUA 520)
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DR. Fuad Al-Awwad
Presented By:
Omar Al-Faris
Majid Al-Suhali

2. TOPICS
 Basic Probability concepts
 Events
 Probability ways
 Law of large numbers
 Round of rule
 Conditional probability distribution
 Random Variables
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3. DEFINITIONS
 Probability: is the likelihood that the event will occur.
 Value of the probability is between 0 and 1
 Sum of the probabilities of all events must be 1.
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4. ROLE OF PROBABILITY IN STATISTICS
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• Probability plays a central role in the important statistical method of hypothesis testing.
Probability VS statistics
•Probability is the field of study that makes statements about what will occur when a sample is drawn
from a known population.
•Statistics is the field of study that describes how samples are to be obtained and how inferences are
to be made about unknown populations.

5. NOTATIONS
 Probability : P(A)
 Statistical experiment: any random activity that result in definite outcome.
 Event: is a collection of one or more outcomes in the statistical experiment.
 Simple event: an event that consists of only ONE outcome, which cannot be broken into simpler components
 Sample space: is the set of ALL simple events.
 The complement of event A, denoted A’ , consists of all events in which event A does not occur.
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6. THE PROBABILITY
 Example 1: Suppose you flip a coin Two times. Find the probability of getting one head and one tail.
 The event is getting one head and one tail.
 There are two simple events that compose this event S=2
 The sample space sets all possible events: {ht,hh,tt,th} so N=4
 The probability of getting one head and one tail is:
P(A)=
number of simple events
number of the sample space
=
2
4
=
1
2
= 0.50%
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7. A- CLASSICAL APPROACH P(A)
 Classical approach determine the indication of the likelihood or probability that a particular outcome will
occur in which that the different simple events are equally likely
 flipping a coin. {head, tail}
 Rolling a dice {1,2,3,4,5,6}
X 1 2 3 4 5 6 sum
p(x) 1/6 1/6 1/6 1/6 1/6 1/6 6/6=1
 Probability of getting 3 children of the same gender.
 It can be displayed by table, tree diagram, or graph 7

8. B- RELATIVE FREQUENCY OF PROBABILITY P(A)
 The Relative Frequency of an event is defined as conduct or observe a procedure and count the number of times
that the event occurs during experimental trials, divided by the total number of trials conducted. 𝑓=
𝑝
𝑡
,
 In which p represents numbers of times A occurred, and t represents number of times the procedure were repeated.
 Based on historical data “estimation”
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9. B- RELATIVE FREQUENCY OF PROBABILITY P(A)
 While watching 10 soccer games where Team 1 plays against Team 2, we record the following final scores:
 What is the relative frequency of Team A winning;? 3 times A won => 3/10= 0.30 = 30% 9
Trial 1 2 3 4 5 6 7 8 9 10
Team A 2 0 1 1 1 1 1 0 5 3
Team B 0 2 2 3 2 1 1 0 0 0

10. C- SUBJECTIVE PROBABILITY
 The probability P (A) of event A is estimated by using knowledge of the relevant
circumstances.
 Quantitative information are gathered are interpreted to help determine this likilhood
through a mathematical mechanism. High flexible, based on judgment. Differs from
person to person.
 Find the probability that David will get hepatitis next year?
 The probability or the chance of team A winning world cup?
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11. HINTS!!!
 THINK CAREFULLY when you try finding the probability value.
 Think about the numbers involved and what they represents, identify the total
number of times being considered.
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12. LAW OF LARGE NUMBERS
 is a theorem that describes the result of repeating the same experiment a large number of times.
 large numbers theorem states that if the same experiment or study is repeated independently a large number of
times, the average of the results of the trials must be close to the expected value. “mean”. The result becomes
closer to the expected value as the number of trials is increased.
 deals only with a large number of trials while the average of the results of the experiment repeated a small number
of times might be substantially different from the expected value.
 The simplest example of the law of large numbers is rolling the dice. The dice involves six different events with
equal probabilities. The expected value of the dice events is:
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13. LAW OF LARGE NUMBERS
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14. COMPLEMENTARY EVENTS
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 The complement of event A, denoted by 𝐴 consists of all outcomes in which A does
not occur.
 Example: in a recent year, there were 3,000,000 skydiving jumps, and 21 of them
resulted in death. Find the probability of not dying when making a skydiving jump.
 P (not dying)=
2,999,979
3,000,000
=0.999993.

15. ROUND OF RULE
If A and B are mutually exclusive events, or those that cannot occur together,
then the third term is 0, and the rule reduces to P(A or B) = P(A) + P(B). For
example, you can't flip a coin and have it come up both heads and tails on one
toss.
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There are three main rules associated with basic probability:
• The Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)

16. ROUND OF RULE
 The Multiplication Rule: P(A and B) = P(A) * P(B|A) or P(B) * P(A|B)
If A and B are independent events, we can reduce the formula to P(A and B) = P(A) * P(B)
Example :
Let use only 50 test results from the subjects who use drugs : (as show in the table)
If 2 subjects are randomly selected with replacement find the probably
that 1st person had positive test result and 2nd person had negative test results.
P(positive test result ) =45/50 P(negative test result)5/50 so,
P(1st subject is positive AND 2nd subject is Negative) = 45/50 *5/50 =0.0900
BUT if same Question without replacement
P(postive test result) = 45/50 P(negative test result) = 5/49 So,
P(1st subject is positive AND 2nd subject is Negative) = 45/50 *5/49 =0.0918 16
45
Positive test results
5
Negative test results
50
Total

17. ROUND OF RULE
 The Complement Rule: P(not A) = 1 - P(A)
For example, if the weatherman says there is a 0.3 chance of rain tomorrow, what are the chances of no
rain?
P(not rain) = 1-P(rain ) = 0.7
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18. CONDITIONAL PROBABILITIES
 A conditional probability of an event is a probability obtained with additional information based of past
results.
 P (B | A) denotes the conditional probability of event B occurring, given that A has already occurred
 Formal approach fro finding P (B | A) =
p( A and B)
p(A)
 For example, the probability that any given person has a cough on any given day may be only 5%. But if
we know or assume that the person has a cold , then they are much more likely to be coughing. The
conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%;
P(Cough | Sick) = 75%
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19. CONDITIONAL PROBABILITIES
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 Refer to the table find
1- if 1 of the 555 randomly selected ,find the probability that the subject had a positive test result ,given that
the subject actually uses drugs .
So, P (B | A) = (positive test result | subject uses drugs)
P (B | A) =
p( subject uses drugs and positive test result)
p(subject uses drugs)
=
45/555
50 /555
= 0.900
Negative test
result
Positive test
result
5
45
Subject uses
drugs
480
25
Subject does not
use drugs

20. RANDOM VARIABLE
 A random variable is defined as the value of the given variable which represents the outcome of a
statistical experiment. It is usually represented by X.
 Example: Tossing a coin: we could get Heads or Tails.
 values Heads=0 and Tails=1 X = {0, 1} , So
We have an experiment (such as tossing a coin)
We give values to each event
The set of values is a Random Variable
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21. RANDOM VARIABLE
 Types :
Random Variables can be either Discrete or Continuous :
 Discrete Data can only take certain values (such as 1,2,3,4,5)
 Continuous Data can take any value within a range (such as a person's height)
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22. QUESTIONS ?
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Thank you