1. Unit 1: Basic Probability

2. Probability:
The chance of occurrence of an event is called the probability and its value always lies between 0 and 1, including 0 and 1. The probability of sure event is always one and probability of an impossible event is always zero.
Method of assigning probability
Classical method: Under this method probabilities are assigned based on laws and rules.
Relative frequency method: under this method probabilities are assigned based on cumulated historical data.
Subjective method: under this method probabilities are assigned based on the feelings or experience or knowledge of person. This probability will be differed by people to people.

3. Basic terms in probability
Experiment:
Event:
Sample space/total events/exhaustive events:
Equally likely events:
Mutually exclusive events vs. not mutually exclusive events:
Independents vs. dependents events:
Favourable events:

4. Experiment:
An experiment is a process which produces outcomes
Event:
The resulting outcomes of an experiment is know as event.
Sample Space/total events/exhaustive events:
The collection of all possible outcomes of an experiment is known as sample space.
Equally likely events:
Two or more events are said to be equally likely events if the chance of occurrence of these events are same.

5. Mutually exclusive vs. not mutually exclusive events:
Two or more events are said to be mutually exclusive events if the they can not occur simultaneously at the same time.
In contrast, if they can occur then they are called not mutually exclusive events.
Independent vs. dependent event:
two or more events are said to be independent event if the occurrence or nonoccurrence of one of the events does not affect the occurrence or nonoccurrence of the other.
In contrast, if the occurrence or nonoccurrence of one of the events affect the occurrence or nonoccurrence of the other then they are called dependent events

6. Mathematical definition of probability:
Favourable events:
Those events which are the interest of our experiment are called favourable events i.e. those events whose probability is to be obtained are called favourable events.
Mathematical definition of probability:
Probability is defied as the ratio of favourable number of events to total number of events. Its value is ranges from 0 to 1. i.e.
Probability =
Favourable number of events
Total number of events

7. Counting Rules mn Rule Sampling from a Population with Replacement
Sampling from a Population without Replacement
Combination
Permutation

8. mn Rule
In a sequence of n events in which the first has k1 possibilities and second event has k2 possibilities and the third has k3 and so forth, the total possibilities of the sequence will be
k1 x k2 x k3 x ………..x kn
Example:
A paint manufacture wishes to manufacture several different paints. The categories include 7 color (Red, blue, white, black, green, brown and yellow) , and 2 use (outdoor and indoor). How many different kinds of paint can be made if a person can select one color and one use?
(Ans: 72 = 14 different paints.)

9. Example:
A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks. A meal consists of one serving of each of the items. How many different meals are available?
(Ans: 548343 = 5,760 meals.)

10. Sampling from a Population with Replacement
Selection of “n” items from a population of “N” items can be done in Nn ways Example: How many different possible sample of size 2 can be selected from population with 4 items Ans: Nn = 42 =16 ways
Sampling from a Population without Replacement
Selection of “n” items from a population of “N” items can be done in
ways
How many different possible sample of size 2 can be selected from population with 4 items
Ans: = = 6 ways

11. Difference between permutation and combination
Combination refer to groups whereas permutation refer to arrangement.
Example of combination:
How many different possible groups of 3 peoples can be formed if they are selected from 5 peoples
Ans:
Example of permutation:
How many different arrangement of 3 peoples are possible if they are selected from 5 peoples.

12. Rules of probability:
Additional rule of probability:
When events are Not Mutually exclusive
When events are mutually exclusive
Multiplication rule of probability
When events are independents
When events are dependents

14. Addition Rule of Probability: Case1: When events are Mutually exclusive

16. Application of types of Probability
Marginal Probability
Union Probability
Joint Probability
Conditional Probability 21

17. Problems 1
From a well shuffled deck of 52 cards, one card is drawn at random: What is the probability that it is
a) Red
b) Spade
c) Ace
d) Red king
e) King or diamond
f) Black or red
g) King and diamond
h) Black and red

18. Problems 2
From a well shuffled deck of 52 cards, two cards are drawn one by one with replacement. What is the probability of getting
a) First card is red and second card is black
b) First is black and second is red
c) Black and red
e) Both are blacks
Problems 3
From a well shuffled deck of 52 cards, two cards are drawn one by one without replacement. What is the probability of getting
a) First card is red and second card is black
b) First is black and second is red
c) Black and red
e) Both are blacks

19. Example 4:
Two fair coins are tossed simultaneously. What is the probability of getting?
a) no head
b) one head and one tail
c) at least one head
d) at most one tail
e) Two head
f) All head
Example 5:
Three fair coins are tossed simultaneously. What is the probability of getting?
a) no head
b) one head and one tail
c) at least one head
d) at most one tail
e) Two head
f) All head

20. Problem 6
When rolling a die once, what is the probability that
a) The face of the die is odd?
b) The face is even or odd?
c) The face is even or one?
d) The face is odd or one?
e) The face is both even and a one?
Problem 7
Two fair dice are thrown at random. What is the probability that the turn up shows
a) a sum 7
b) sum of 8 or 9
c) a sum less than 5
d) the two dice show the same number
e) the number on the two dice differ by more than 2
f) the product of the two number is even.

21. Problem 8
One ball is drawn at random from a bag containing 4 red, 5 white and 6 blue balls. What is the probability that it is
a) Red
b) White
c) red or white
d) white or green
e) red or white or green
Problem 9
A bag contains 7 red balls and 5 white balls. 4 balls are drawn at random, what is the probability that
a) All of them are red
b) Two of them are red and two white
c) At least one red
d) At most two red

22. Problems 10:
Five men in a company of 20 are graduates. If 3 men are picked out from 20 at random, what is the probability that they are
a) all graduates
b) all non-graduates
c) 2 graduates
d) at most 1 graduates
e) at least one graduates
f) 2 non graduates

23. Problem:
A sample of 500 respondents was selected in a large metropolitan area in order to determine various information concerning consumer behaviors. Among the question asked was” do you enjoy shopping for clothing?" Of 240 males, 136 answered yes. Of 260 females, 224 answered yes. What is the probability that a respondent chosen at random
a) is a male?
b) enjoys shopping for clothing?
c) is females?
d) is females and enjoys shopping for clothing?
e) is male and does not enjoys shopping for clothing?
f) is females or enjoys shopping for clothing?
g) is male or females?
h) Suppose the respondent chosen is a female. What then is the probability that she does not enjoy shopping for clothing?
i) Suppose the respondent chosen enjoys shopping for clothing. What then is the probability that the individual is a male?

24. Application of Conditional Probability

25. Problem 15:
If P (A and B) =0.4 and P (B) =0.8, find P (A/B)
Problem 16
If P (A) =0.7 and P (B) =0.6, and if A and B are statistically independent, find P (A and B)
Problem 17
Given that P(A) = 3/14, P(B) = 1/6, P(C) = 1/3 ,P(AC) = 1/7 and P(B/C) = 5/21. Find the following probabilities:
a) P(A/C)
b) P(C/A)
c) P(BC)
d) P(C/B)

26. Problem
The southeast regional manager of General Express, a private parcel delivery firm is worried about the likelihood of strikes by some of his employees. He has learned that the probability of a strike by his pilots is 0.75 and the probability of a strike by his drivers is Further, he knows that if the drivers strike, there is a 90 percent chance that the pilots will strike in sympathy.
a) What is the probability of both groups striking? (0.585)
b) If the pilots strike, what is the probability that the drivers will strike in sympathy? (0.78)

27. Application of Bayes’ Theorem

28. Problems 21
A Company has two plants to manufacture the screws. Out of screws, plant I manufactures 80% of the screws and plant II manufactures 20%. At plant I, 85% screws are rated standard quality. At plant II, only 65% screws are rated standard quality. One screw is selected at random from the whole consignment and was found standard quality. What is the probability that the selected screw was
Manufactured by Plant I
Manufactured by Plant II