PROBABILITY THEORY

Oyindamola Bidemi Yusuf

What is Probability?

· Measurement of uncertainty· Theory of choice and chance · Allows intelligence guess about future · Helps to quantify risk· Predicts outcomes

PROBABILITY DEFINITION-OBJECTIVE

Frequency Concept· Based on empirical observations· Number of times an event occurs in a

long series of trials

PROBABILITY DEFINITION - SUBJECTIVE

Merely expresses degree of belief

Based on personal experience

Basic Terminologies

Experiment(Process of conducting trials)Trial (Act of an experiment.)Outcome ( Result of a Particular

trial)Event (Particular outcome or

single result of an experiment)

PROBABILITY CALCULATIONS

CLASSICAL PROBABILITY

Classical Probability

Count number favorable to event E = a · Count number unfavorable to event E = b· Total favorable and unfavorable = a+b· Assume E can occur in n possible ways· Assume occurrence of events equally likely· Total number of possible ways =a+b = n

Probability of an event-E

Probability of E = a = Pr(E)

a+b

Number of times event favorable divided by number of all possible ways.

Probability Thermometer

. 1.0 - sure to occur

– - 0.5

0- cannot occur

0>Pr (E) < 1

Type of Events

· Simple events· Compound events· Mutually exclusive events· Independent events

Simple events

Events with single outcomes

tossing a fair coin

Compound events

Compound events is the combination of two or more than two simple events.

Suppose two coins are tossed simultaneously

Probability Rules

Addition rule

Multiplication rule

Addition rule

Single 6-sided die is rolled.

What is the probability of rolling a 2 or a 5?  

P(2)  =  1/6

P(5)  =  1/6

P(2 or 5)  =  P(2)  +  P(5)    = 1/6  +  1/6 =2/6

Mutually Exclusive Events

Pr (A or B) = Pr (A) + Pr (B)

IF not mutually exclusive Pr (A or B) = Pr (A) + Pr (B) - Pr (A and B)

QUESTION ON MUTUALLY EXLUSIVE EVENTS

From the records at an STC, 4 girls had HIV, 4 other girls had gonorrhea while 2 girls have both gonorrhea and HIV.

What is the probability that any girl selected will have

i. HIV only

ii. HIV or Gonorrhea.

SOLUTION

Prob. Of HIV only =4/10

– Prob. of HIV or Gonorrhea = Pr(HIV) +

Prob.(Gonorrhea) - Pr(HIV and Gonorrhea) = 4/10 + 4/10 - 2/ 10

Independent events

Choosing a marble from a jar AND landing on heads after tossing a coin.

Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.

Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.

Multiplication Rule

When two events, A and B, are independent, the probability of both occurring is:  

P(A and B) = P(A) · P(B)

A coin is tossed and a single 6-sided die is rolled.

Find the probability of landing on the head side of the coin and rolling a 3 on the die. P(head)  =  1/2

P(3)  =  1/6

P(head & 3)  =  P(head)  ·  P(3)    

=1/2 x 1/6  =  1/12

Conditional Probability

In probability theory, a conditional probability is the probability that an event will occur, when another event is known to occur or to have occurred.

Conditional Probability

Events not independent

· Pr (A given B) = Pr (A and B)

Pr (B)

On the “Information for the Patient” label of a certain antidepressant, it is claimed that based on some clinical trials, there is a 14% chance of experiencing sleeping problems known as insomnia (denote this event by I),

26% chance of experiencing headache (denote this event by H), and there is a 5% chance of experiencing both side effects (I and H).

Suppose that the patient experiences insomnia; what is the probability that the patient will also experience headache?

Since we know (or it is given) that the patient experienced insomnia, we are looking for P(H | I). According to the definition of conditional probability:

P(H | I) = P(H and I) / P(I) = 0.05/0.14 = 0.357.

Random Variables

A real valued function, defined over the sample space of a random experiment is called the random variable, associated to that random experiment.

That is the values of the random variable correspond to the outcomes of the random experiment.

Random Variables

Take specified values with specified probabilities

Discrete Random variable–E.g. no of children in a family, no of

patients in a doctors surgeryContinuous Random variable

The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values.

It is also sometimes called the probability function or the probability mass function

Continuous random variables are usually measurements.

Examples include height, weight, the amount of sugar in an orange, the time required to run a mile, etc

DISCRETE PROBABILITY DISTRIBUTION

Binomial

BINOMIAL DISTRIBUTION

Successive trials are independent

Only two outcomes are possible in

each trial or observation

Chance of success in each trial is known

Same chance of success from trial to trial

BINOMIAL FORMULA

Pr (r out of n events) = n ! pr qn-r

r! (n-r) ! where n ! =n(n-1)(n-2)(n-3)….2.1

e.g. 3! =3x2x1

BINOMINAL TERMS

N = Number of trials

r = Number of successes

p = Probability of success in each trial

q = 1-p = Probability of failure in each trial

! = Factorial sign

EXAMPLE BINOMINAL

It is known that 10% of patients diagnosed to have a condition survive following surgical treatment. What is the chance of 2 people surviving out of 5 diagnosed with the condition and treated surgically.

Solution

Continuous Probability Distribution

NORMAL

The Normal Curve

The Shape of a Distribution

Symmetrical– can be divided at the center so that each half

is a mirror image of the other

Asymmetrical

Skewness

– If a distribution is asymmetric it is either positively skewed or negatively skewed.

– A distribution is said to be positively skewed if the values tend to cluster toward the lower end of the scale (that is, the smaller numbers) with increasingly fewer values at the upper end of the scale (that is, the larger numbers).

With a negatively skewed distribution, most of the values tend to occur toward the upper end of the scale while increasingly fewer values occur toward the lower end.

Negative Skewness

Properties of Normal Curve

Bell shaped and symmetric about centre

Completely determined by its mean and standard deviation

Mean, median and mode have same value

Total area under curve is 1 (100%).

68% of all observations lie within one standard deviations of the mean.

95% of observations lie within 1.96 standard deviations of the mean value

Gives probability of falling within interval if data has

normal distribution.

Importance of Normal Distribution

Fits many practical distributions of variables in medicine

If variables are not normally distributed, transformation techniques to make them normal exist.

Sampling distributions of means and proportions are known to have normal distributions

It is the cornerstone of all parametric tests of statistical significance.

Presentation of Normal Distribution.

As a mathematical equationGraphTable

-

1. Mathematical Equation

- 1/ 2 (x - )2

y = 1___ e

2II

II and e are constants

is arithmetic mean

is standard deviation

Normal distribution curve

The Standardized normal distribution

All normal distributions have same overall shape

Peak and spread may be different

However markers of 68th and 95Th percentiles will still be located at 1 and 2 SD

This attribute allows for standardization of any normal distribution

Can define distance along x axis in terms of SD from the mean instead of the true data point

Condenses all normal distributions into one through a mathematical equation

Z= x- μ

σ

Each data point is converted into a standardized value, and its new value is called a Z score

Z Score

Standardizing data on one scale so that a comparison can be made

Standard score or Z score is:– The number of standard deviations from the

mean

convert a value to a Standard Score: – first subtract the mean, – then divide by the Standard Deviation

Z Score

The z-score is associated with the normal distribution and it is a number that may be used to: – tell you where a score lies compared with the

rest of the data, above/below mean. – compare scores from different normal

distributions

Table of Area

Areas under a standard normal curveGives probability of falling within an interval.Standard normal curve has a mean = 0 and standard deviation = 1 Need to transform data to standard normal curve to use this table.

1. Transformation to standard Normal Curve.

- Use Z = (x - )

Z is standardized normal deviate or normal

score.

- Read corresponding area from table.

- Z is in the Ist column in the table.

- Area in the heart of the table.

The IQs of a group of students are normally distributed with a mean of 100 and a standard deviation of 12. What percentage of students will have an IQ of 110 or more?

Z= x- μ/ σ

Z= (110-100)/12

Z=0.83, this corresponds to 0.2033 from the table

20% of students will have an IQ of 100 or more

What % of students will have IQ between 100 and 110?

If the heights of a population of men are approximately normally distributed with mean of 172m and standard deviation of 6.7cm. What proportion of men would have heights above 180cm.

solution

Z= x- µ

σ

180-172 = 1.19

6.7

In the table of normal distribution, the probability of obtaining a standardised normal deviate greater than 1.19 is 0.117(11.7%)

Therefore around 12% of the population would have heights above 180cm.

In summary

Normal distribution as a predictor of events

Direct applications in statistics

Testing for significance

Backbone of inferential statistics

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