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Probability as a general concept can be defined as the chance

of an event occurring.

Probability theory is used in the field of insurance,

investment, weather forecasting and various other areas.

The concept of probability is frequently encountered in

everyday communication. Example, A physician.

Most people express probabilities in terms ofpercentages.

But, it is more convenient to express probabilities asfractions. Thus, we may measure the probability of the

occurrence of some event by a number between 0 and 1.


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Experiment a situation involving chance or

probability that leads to results.

Outcomes - the result of a single trial of an

experiment.

Sample space The collection of all possible

outcomes for an experiment.


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For example, in the rolling of the die, each of the six sides is equally

likely to be observed. So, the probability that a 4 will be observed isequal to 1/6.


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It is an approximation to the true probability of an event.However, if we were able to perform our process more and moretimes, the relative frequency will eventually approach the actualprobability.

P(E) = m/N.The example : If we tossed the two dice 100 times, 200 times,300 times, and so on, we would observe that the proportion of

6's would eventually settle down to the true probability.

an individual's personal judgment about whether a specific

outcome is likely to occur. Subjective probabilities contain noformal calculations and only reflect the subject's opinions andpast experience.The example: A Rangers supporter might say, "I believe thatRangers have probability of 0.9 of winning the Scottish Premier

Division this year since they have been playing really well."


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Given some process (or experiment) with n mutually exclusiveevents E1, E2, , En, then

1- P (Ei) 0, i = 1, 2, n2- P (E1) + P (E2) + + P (En) = 1

A B means A or B.

Let S = {1,2,3,4,5,6,7,8,9,10},A be choosing an odd number > 2,then A = {3,5,7,9}, P(A) = 0.4 andB be choosing a number divisible by 3,then B = {3,6,9}, P(B) = 0.3.

A B = {3,5,6,7,9} and P(A B) = 0.5.

A

B A

63

9

5

7

1 2

4 810


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A B means A and B.

In the above example, A = {3,5,7,9},

and B = {3,6,9}, then

A B = {3,9} and P(A B) = 0.2.

A

B

A

63

9

5

7

1 2

4 810

A` means the complement of A, where

A A` = S and A A` =.

In the above example,B = {3,6,9}, P(B) = 0.3, thenB` = {1,2,4,5,7,8,10} and P(B`) = 0.7.

1 2

4 8

6 5

7

3

9

10

A

B

B


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1- A and B are called disjoint if A B = , and then P(A B) = 0 and P(A B) = P(A) + P(B).

if A is choosing an odd number < 11,A = {1,3,5,7,9} andB is choosing an even number < 11,B = {2,4,6,8,10}.

Then P(A B) = 0 and P(A B) = P(A) + P(B) = 1.

2- If A and B are not disjoint, thenP(A B) = P(A) + P(B) - P(A B)

if A is choosing a number divisible by 5A = {5,10} andB is choosing an even number < 11,B = {2,4,6,8,10}.

Then P(A B) = 0.1 andP(A B) = P(A) + P(B) - P(A B) = 0.6.

2 4 6 8 101 3 5 7 9

A B

10 2 4 6 8

BA

51

3

7 9


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Here is the data of a sample of people in a certain city:

SumFemale

(B`)

Male

(B)

23815Diabetic

(A)1026240Normal

(A`)1257055Sum

P(A) = 23 / 125P(A`) = 1- P(A)

= 102 / 125P(B) = 55 / 125P(B`) = 70 / 125

P(A B) = 15 / 125P(A B`) = 8 / 125P(A` B) = 40 / 125P(A` B`) = 62 / 125

P(A) = P(A B) + P(A B`)

= 23 / 125P(A B) = P(A) + P(B) - P(A B)

=( 23 / 125) + (55 / 125) (15 / 125)

= 63/12515 840

62

15

AB


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Events are mutually exclusive if they cannot

happen at the same time.

For example, if we toss a coin, either heads or

tails might turn up, but not heads and tails at

the same time.

Similarly, in a single throw of a die, we can

only have one number shown at the top face.

The numbers on the face are mutually

exclusive events


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IfA and B are mutually exclusiveevents then the probability ofA

happening OR the probability ofB

happening is P(A) + P(B).

P(A or B) = P(A) + P(B)


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Example

What is the probability of a die showing a 2 or

a 5?


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Probability of an event or outcome based on the occurrence of aprevious event or outcome. Conditional probability is

calculated by multiplying the probability of the preceding

event by the updated probability of the succeeding event.

If A and B are events with P(A) 0, then the

conditional probability of B given A is defined by


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Example

If a coin is tossed 3 times, what is the probability that all three

tosses come up heads given that at least two of the tosses come

up heads? Conditional probability of

Solution:


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Events are independent if the outcome of oneevent does not affect the outcome of another.For example, if you throw a die and a coin, the

number on the die does not affect whetherthe result you get on the coin.

IfA and B are independent events, then theprobability ofA happening AND theprobability ofB happening is P(A) P(B).

P(A and B) = P(A) P(B)


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Example

If a dice is thrown twice, find the probability

of getting two 5s.


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