Probability – CS Foundation Statistics Notes

1. Probability
Probability or likelihood is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen) Probability theory is a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

The fundamental ingredient of probability theory is an experiment that can be repeated, at least hypothetically, under essentially identical conditions and that may lead to different outcomes on different trials. The set of all possible outcomes of an experiment is called a sample space. For example, individuals in a population favoring a particular candidate in an election may be identified with balls of a particular color, those favoring a different candidate may be identified with a different color, and so on. Probability theory provides the basis for learning about the contents of the urn from the sample of balls drawn from the urn; an application is to learn about the electoral preferences of a population on the basis of a sample drawn from that population.

2. Basic concepts of set theory
The word ‘set’ in mathematics was first of all used by George Cantor. According to him, ‘A set is any collection into a whole of definite and distinct objects of our intuition or thought’. However, Cantor’s definition faced controversies due to the forms like ‘definite’ and ‘collection into a whole’.

3. Set
A set is a collection of objects which are called the members or elements of that set. If we have a set we say that some objects belong (or do not belong) to this set, are (or are not) in the set. We say also that sets consist of their elements.

Examples: the set of students in this room; the English alphabet may be viewed as the set of letters of the English language; the set of natural numbers 1; etc.
For example: is the letter q the same thing as the letter Q? Well, it depends on what set we are considering. If we take the set of the 26 letters of the English alphabet, then q and Q are the same elements. If we take the set of 52 upper-case and lower-case letters of the English alphabet, then q and Q are two distinct elements. Either is possible, but we have to make it clear what set we are talking about, so that we know whether or not q = Q.
Sets are to be distinct and well defined.
Subset
Subset and Proper Subset
Set A is a subset of set B if every element of A is also an element of B.
Set A is a proper subset of B if every element of A is also an element of B, but A CANNOT be exactly the same as B.

For example:
A = {a,b,c,d,e}
B = {a,b,c,d,e,f}
A is said to be a subset and a proper subset of B.
The number of subsets of a set with n elements is 2n.
For example: If a set has 5 elements, it will have 25 or 32 subsets.
The number of proper subsets of a set with n elements is 2n – 1.
For example: If a set has 5 elements, it will have 25 – 1 or 31 proper subsets.

4. Union of Sets
The union of sets A and B, written as AUB, is the set of elements that appear in either A or B.
For example:
A ={1,2,3,4,5}
B= {2,4,6,8,10}
The union of A and B (i.e. AUB) is {1, 2, 3, 4, 5,

5. Difference of Sets
The difference between sets A and B, written as A-B, is the Set of elements belonging to set A and NOT to set B.
For example:
A ={1,2,3,4,5}
B = {2,3,5}
The difference of A and B (i.e. A-B) is {1,4}.
A-B ≠ B-A

6. The Universal Set
In some problems involving sets, it is necessary to consider one or more sets under consideration as belonging to some larger set that contains them.
For example
If we were considering the set of skilled workers (S, say) on a production line, it might be convenient to consider the universal set (U, say) as all of the workers on the line. In other words, where a universal set has been defined, all the sets under consideration must necessarily be subsets of it.

7. The complement of a set
If A is any set, with some universal set U defined, the complement of A, normally written as A’, is defined as ‘all those elements that are not contained in A but are contained in U’.
For an example of the workers on the production line (given above), S was specified as the set of skilled workers within the universal set of all workers on the line. Therefore, S’ would be all the workers that were not skilled, i.e. the set of unskilled workers.

8. Set Intersection
The intersection of two sets A and B is written as AC B and defined as that set that contains all the elements lying within both A or B.
For example, if A = (a,b,c,d,f,g,) and B = (c,f,g,h,j), then the intersection of A and B is A << B = (c,f,g), since these are the elements that lie in both sets. The intersection of three or more sets is a natural extension of the above. If P, Q, and R are any three sets then PC QC R is the set containing all the elements that lie in all three sets. Any combinations of union and intersection can be used with sets. For, example, if X and Y are the sets specified above and Z = (d,f,g,j). then: (XCy) >>Z = (c,f,g) >>(d,f,gj) =(c,d,f,gj) which can be described in words as ‘the set of elements that are in either both of X and Y or in Z’.

9. Factorial
The product of a given positive integer multiplied by all lesser positive integers: The quantity four factorial (4!) =4 • 3 • 2 • 1 = 24. Symbol: n! where n is the given integer.

10. Factorials are a recursive function
n! = n*(n-1)! Where 0! = 1
So l! = 1*1
2! =2*1*1=2
3!=3*2*1*1=6
4! =4*3*2*1*1=24
etc.
Factorials often come up in probability because of the way permutations work.

For example
If you pick a single card out of a deck, there are 52 different ways for your card drawing to turn out. Any of the 52 cards could be drawn. If you now pick a second card without putting the first back, there are 51 ways for that second card to be drawn (Any of the 51 remaining cards) So the amount of different combinations of card picks that could happen would be 52*51.

You can then see that if you were to draw every card in the deck, there would be 52*51*50*……. = 521 ways for you to draw them. This means that if you draw 52 cards, then the order that you drew them in had a 1/(52!) chance of occurring, a very small chance (Factorial is one of the fastest-growing functions that’s often used). Many situations in probability can be likened to card decks, so you might be able to imagine that factorials can come up in all sorts of applications for probability.

11. Experiment
It is an operation that has two or more results and performing the experiment is called a trial.

  • Probability experiment

An action through which specific results (counts, measurements, or responses) are obtained.

  • Outcome

The result of a single trial in a probability experiment.

  • Sample Space

The set of all possible outcomes of a probability experiment

  • Event

One or more outcomes and is a subset of the sample space

  • Simple event

A simple event is the number of occurrences divided by the number of possible occurrences.

  • Compound event

Compound events are the combination of multiple simple events, such as rolling two dice

  • Mutually Exclusive

Mutually Exclusive means you can’t get both events at the same time. It is either one or the other, but not both

Examples:
Turning left or right are Mutually Exclusive (you can’t do both at the same time)
Heads and Tails are Mutually Exclusive
Kings and Aces are Mutually Exclusive

Equally likely events
Two or more events that have an equal probability of occurrence are said to be equally likely, i.e. if on taking into account all the conditions, there should be no reason to accept any one of the events in preference over the others.

Examples
First example
A : The event of getting a “HEAD” and
B: the event of getting a “TAIL”
Events “A” and “B” are said to be equally likely events Both the events have the same chance of occurrence.
2-second example Where
A: The event of getting 1
B: the event of getting 2…….
F: the event of getting 6
Events “A”, “B”, “C”, “D”, “E”, “F” are said to be equally likely events All these events have the same chance of occurrence.

• Exhaustive events
One or more events are said to be exhaustive if all the possible elementary events under the experiment are covered by the events considered together. In other words, the events are said to be exhaustive when they are such that at least one of the events compulsorily occurs. Exhaustive events maybe elementary or compound events. They may be equally likely or not Equally likely.

12. Mathematical probability
The probability of an event consisting of n out of m possible equally likely occurrences, defined to be a mathematical probability. It is also known as classical probability.
The number of outcomes favorable to the occurrence of an event is divided by the total number of possible outcomes. In order for this ratio to be valid, each of the outcomes must be equally likely. Distributions are gained from actual occurrences in long-run experience and experimentation. An example is repeated trials under a constant-cause situation.
Probability of an event = m/m+n
Where m= number of outcomes in favor of event
m+n=total number of equally likely mutually exclusive and exhaustive events
The probability of rolling a six on a single roll of a die is 1/6 because there is only 1 way to roll six out of 6 ways it could be rolled. The probability of getting a sum of 5 when rolling two dice is 4/36 = 1/9 because there are 4 ways to get a five and there are 36 ways to roll the dice (Fundamental Counting Principle – 6 ways to roll the first times 6 ways to roll the second).

Do not make the mistake of saying that the probability of rolling a sum of 5 is 1/11 because there is one 5 out of a sample space of 11 sums (2 through 12). When the sample spaces are not equally likely, do not divide by the number in the sample space.

13. Properties of Probabilities
All probabilities are between 0 and 1 inclusive.
A probability of 0 means an event is impossible, it cannot happen.
A probability of 1 means an event is certain to happen, it must happen.
Example out of the below mentioned a) and c) cannot be a probability because A probability is always greater than or equal to 0 and less than or equal to 1
(a) -0.00001
(b) 0.5
(c) 1.001
(d) 0
(e) 1
Example A die is rolled, find the probability that an even number is obtained.

Solution
Let us first write the sample space (total number of cases) M of the experiment. M= {1,2,3,4,5,6}
Let A be the event “an even number is obtained” and write it down.
A ={2,4,6}
We now use the formula of the classical probability.
P(A) = n(A)/M
= 3/6
= 1/2

Example
Two dice are rolled, find the probability that the sum is
(a) equal to 1
(b) equal to 4
(c) less than 13 Solution

(a) The sample space S (total number of events) of two dice is shown below.
S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2.1) ,(2,2),(2,3),(2,4),(2,5),(2,6)
(3.1) ,(3,2),(3,3),(3,4),(3,5),(3,6)
(4.1) ,(4,2),(4,3),(4,4),(4,5),(4,6)
(5.1) ,(5,2),(5,3),(5,4),(5,5),(5,6)
(6.1) ,(6,2),(6,3),(6,4),(6,5),(6,6)}
Let E be the event “sum equal to 1”. There are no outcomes which correspond to a sum equal to 1, hence P(E) = n(E)/n(S) = 0/36 = 0
(b) Three possible outcomes give a sum equal to 4: E = {(1,3),(2,2),(3,1)}, hence. P(E) = n(E)/n(S) = 3/36 = 1/12
(c) All possible outcomes, E = S, give a sum less than 13, hence. P(E) = n(E)/n(S) = 36/36 = 1

14. Addition Rules of probability
When you want to find the probability of one event OR another occurring, you add their probabilities together. To find the probability of event A or B, we must first determine whether the events are mutually exclusive or non-mutually exclusive. Then we can apply the appropriate

15. Addition Rule

Addition Rule 1
When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event P(A or B) = P(A) + P(B)

Addition Rule 2
When two events, A and B, are non-mutually exclusive, there is some overlap between these events. The probability that A or B will occur is the sum of the probability of each event, minus the probability of the overlap.
P(A or B) = P(A) + P(B) – P(A and B)
Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the
the sum of the probability of each event.
P(A or B) = P(A) + P(B)
A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?
Let’s use this addition rule to find the probability

Probabilities: P(2) = \(\frac{1}{6}\)
P(5) = \(\frac{1}{6}\)
P (2 or 5) = P (2) + P (5)
= \(\frac{1}{6}\) + \(\frac{1}{6}\)
= \(\frac{2}{6}\)
= \(\frac{1}{3}\)

Let’s look at some experiments in which the events are non-mutually exclusive.
Example: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club?
Probabilities: P(king or club) = P(king) + P(club) – P(king of clubs)
= \(\frac{4}{52}\) + \(\frac{13}{52}\) – \(\frac{1}{52}\) = \(\frac{16}{52}\) = \(\frac{16}{52}\) = \(\frac{4}{13}\)

In Example, the events are non-mutually exclusive. The addition causes the king of clubs to be counted twice, so its probability must be subtracted. When two events are non-mutually exclusive, a different addition rule must be used.

Addition Rule 2: When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:
P(A or B) = P(A) + P(B) – P(A and B)
In the rule above, P(A and B) refers to the overlap of the two events

16. Multiplication Rules
When you want to find the probability of two events that are jointly occurring, then you need to apply the multiplication rule. This principle can be extended to probabilities.

-Independent Events
Independent Events are events where one occurring doesn’t change the probability of the other occurring. When events are independent, the probability of them both occurring is…
P(A and B) = P(A) * P(B)
We don’t have time to get into probability very deeply. If we did, we would cover conditional

17. Probability – the probability of dependent events.
Suppose we roll one die followed by another and want to find the probability of rolling a 4 on the first die and rolling an even number on the second die. Notice in this problem we are not dealing with the sum of both dice. We are only dealing with the probability of 4 on one die only and then, as a separate event, the probability of an even number on one die only.
P(4) = 1/6
P(even) = 3/6
So P(4 ∩ even) = (l/6)(3/6) = 3/36 = 1/12

18. Complementary Events
The root word in complementary is “complete”. Complementary events complete, or make whole. Complementary events are mutually exclusive, but when combined make the entire sample space. The symbol for the complement of event A is A’. Some books will put a bar over the set to indicate its complement. Since complementary events are mutually exclusive, we can use the special addition rule to find their probability. Furthermore, complementary events are all-inclusive, *so they make the sample space when combined, so their probabilities have a sum of 1.

The sum of the probabilities of complementary events is 1.
P(A) + P(A’) = 1
P(A’) = 1 – P(A)
Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. We are going to pull out the first marble, leave it out, and then pull out another marble. What is the probability of pulling out a red marble followed by a blue marble?

We can still use the multiplication rule which says we need to find P(red) • P(blue). But be aware that in this case when we go to pull out the second marble, there will only be 8 marbles left in the bag.
P(red) = 2/9
P(blue) = 3/8
P(red ∩ blue) = (2/9)(3/8) = 6/72= 1/12

19. Random variable
A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random, phenomenon. There are two types of random variables, discrete and continuous.

20. Discrete Random Variables
A discrete random variable is one that may take on only a countable number of distinct values such as 0,1,2,3,4, Discrete random variables are usually (but not necessarily) counts. For a discrete random variable, its probability distribution is any table, graph, or formula that gives each possible value and the probability of that value. It is also called the probability distribution function.

Example
Consider an experiment where a coin is tossed three times. If X represents the number of times that the coin comes up heads, then X is a discrete random variable that can only have the values 0,1,2,3 (from no heads in three successive coin tosses to all heads). No other value is possible for X.

21. Continuous Random Variable
A continuous random variable is one that takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile. Consider an experiment where a coin is tossed three times. If X represents the number of times that the coin comes up heads, then X is a discrete random variable that can only have the values 0,1,2,3 (from no heads in three successive coin tosses to all heads). No other value is possible for X.

An example of a continuous random variable would be an experiment that involves measuring the amount of rainfall in a city over a year or the average height of a random group of 25 people.

22. Probability Distribution
The probability distribution of a 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, An example will make clear the relationship between random variables and probability distributions. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.

A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example if a probability distribution.

Number of heads Probability
0 0.25
1 0.50
2 0.25

The above table represents the probability distribution of the random variable X.
Types of the probability distribution

  • Discrete probability distributions- The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It shows the following properties. Firstly the probability of a discrete random variable is between 0 to 1 and secondly, the sum of all properties is 1.
  • Continuous probability distributions- The number of possible values in a range is infinite in the case of a continuous probability distribution. The probability function is determined by the area under the graph.

23. Expected value
Expected value is a concept employed in statistical analysis. It is a weighted average approach that involves multiplying each possible outcome in a situation with its probability to arrive at the expected outcome. Thus Expected value is a measurement of the center of a probability distribution.
Example
Flip a coin three times and let X be the random variable of the number of heads. This has probability distribution of 1/8 forX= 0, 3/8 forX= 1, 3/8 for A= 2, 1/8 forX= 3. Use the expected value formula to obtain: (1/8)0 + (3/8)1 + (3/8)2 + (1/8)3 = 12/8 =1.5

Probability MCQ Questions

Question 1.
Probability is
a. the study of the randomness of happening of an event
b. the haphazard happening of an event
c. mathematical study of the randomness of happening of an event
d. the mathematical study of a haphazard happening
Answer:
c. mathematical study of the randomness of happening of an event

Question 2.
If two sets are denoted as ACB it shows that
a. B is proper to set of set A
b. B is smaller and A is a set and A is a set of B
c. A is smaller than B and A is a proper sub-set of B
d. None of me above
Answer:
c. A is smaller than B and A is the proper subset of B

Question 3.
If it is shown as B>A it shows that
a. B is a Subset of A
b. A is Subset B
c. B is a superset of A
d. A is a superset of B
Answer:
c. B is a superset of A

Question 4.
Equal sets in probability mean
a. Element in both the sets are equal
b. Element in one set is monthly equal to the elements in other set
c. both a & b
d. None of the above
Answer:
a. Element in both the sets are equal

Question 5.
We can depict the two-set union as
a. A = B
b. A + B
c. A Ω B
d. A ∪ B
Answer:
d. A ∪ B

Question 6.
The intersection of sets is denoted as
a. A ∪ B
b. A Ω B
c. A > B
d. A< B
Answer:
b. A Ω B

Question 7.
There are 600 electric bulbs in a box out of which 20 bulbs are defection if one bulb is chosen at random from the box. What is here probability that the chosen bulb is defective?
a. 1/19
b. 1/30
c. 1/25
d. 1/16
Answer:
b. 1/30

Question 8.
The probability of a sure event is
a. 1
b. 0
c. Nil
d. unknown
Answer:
a. 1

Question 9.
The most appropriate definition of Experiment is
a. where two or more outcomes are received when a trial is alone
b. depilate one outcome is received when a trial is done
c. Both a & b
d. alone of the above
Answer:
a. where two or more outcomes are received when a trial is alone

Question 10.
Sample space is also called
a. universal space
b. event space
c. possibility space
d. All of the above
Answer:
b. event space

Question 11.
If E be an event then P(E) + (None)=
a. 0
b. 1
c. 2
d. None of the above
Answer:
b. event space

Question 12.
When a choice is a thousand then possible events are
a. 1
b. o
c. 6
d. 4
Answer:
c. 6

Question 13.
A coin is tossed 60 times with the following result head -28 times Tail -32 times Find the probability of getting head if the coin is thrown once
a. 1
b. 7/15
c. 2
d. none of the above
Answer:
b. 7/15

Question 14.
The probability of occurrence of any limit lies between
a. 0 and 1
b. 0 and 2
c. 0 and 3
d. 0 and infinite
Answer:
a. 0 and 1

Question 15.
Each outcome of the sample space is called
a. event space
b. sample point
c. possibility point
d. universal point
Answer:
b. sample point

Question 16.
If E is an event, then <P(E)< is
a. 0
b. 1
c. both a & b
d. Indefinite
Answer:
c. both a & b

Question 17.
Probability can be
a. measured numerically
b. Can not be measured numerically
c. Can be measured but not necessarily nominally
d. All Of the above
Answer:
a. measured numerically

Question 18.
A box contains 3 red balls 4 white balls and 7 black balls one ball is chosen at random what the probability of choosing the black ball is.
a. 2/5
b. 2/4
c. 1/3
d. 5/7
Answer:
d. 5/7

Question 19.
If the events in the probability are independent then
a. The occurrence of one event does not affect the occurrence of record
b. The occurrence of one event still affects the occurrence event
c. It is innovative
d. Both a & b
Answer:
a. The occurrence of one event does not affect the occurrence of record

Question 20.
How many types of random variables are there
a. 2
b. 3
c. O
d. 4
Answer:
a. 2

Question 21.
Weight, lengths are examples of typing of _______ random variable
a. Concrete random variable
b. continuous random variable
c. Both a & b
d. None of the above
Answer:
b. continuous random variable

Question 22.
It is given that the probability of winning a game is
0. 7 what is the probability of losing the game?
a. 3
b. 7
c. 7
d. 4
Answer:
a. 3

Question 23.
Suppose you toss a coin twice what is the probability of tossing two heads?
a. 1/16
b. 1/4
c. 1/8
d. 1/5
Answer:
b. 1/4

Question 24.
Suppose you toss a coin twice. What is the probability of tossing two heads given that your first toss is ahead?
a. 1/16
b. 1/8
c. 1/4
d. 1/2
Answer:
d. 1/2

Question 25.
The collection of all possible events is called
a. A probability
b. A sample space
c. An event
d. Random variable
Answer:
b. A sample space

Question 26.
The term ______ probability of A and B is used to denote the probability of the intersection of A and B
a. Marginal
b. Conditional
c. Subjective
d. Joint
Answer:
d. Joint

Question 27.
There is a bag filled with 8 balls 3 white and 5 black. If ball is drawn from the bag without replacement the probability that the second ball drawn is black is, given that the first ball drawn is black is
a. 2/7
b. 5/7
c. 4/7
d. None of the above
Answer:
c. 4/7

Question 28.
Cholesterol measurements constitute
a. Continuous random variable
b. Discrete random variable
c. Qualitative random variable
d. None of the above
Answer:
a. Continuous random variable

Question 29.
Consider two events A & B. Event A has a probability of 1/2 while event B has a probability of 1/20 then
a. A is less probable to occur
b. A has more probability to occur
c. At least 20 trails are needed for B to appear
d. None of the above
Answer:
b. A has more probability to occur

Question 30.
Suppose you draw one card from a standard 52 card deck. What is the probability that the card is black and a jack?
a. 1/26
b. 1/64
c. 1/52
d. 1/25
Answer:
d. 1/25

Question 31.
The blood groups of 200 people are distributed as follows: 50 have type A blood, 65 have a B blood type, 70 have the O blood type and 15 have type AB blood. If a person from this group is selected at random, what is the probability that this person has an O blood type?
a. 0.35
b. 0.25
c. 0.45
d. none of the above
Answer:
a. 0.35

Question 32.
A die is rolled, find the probability that the number obtained is greater than 4.
a. 1/3
b. 2/3
c. 1/4
d. 2/4
Answer:
a. 1/3

Question 33.
Two coins are tossed, find the probability that one head only is obtained
a. 1/3
b. 2/3
c. 1/2
d. 2/4
Answer:
c. 1/2

Question 34.
Two dice are rolled, find the probability that the sum is equal to 5.
a. 1/9
b. 2/3
c. 1/6
d. 2/4
Answer:
a. 1/9

Question 35.
If all the elements of set A belong to set B and all the elements of set B belong to set A. This set is called.
a. Equal set
b. Disjoint set
c. Null Set
d. Void set
Answer:
a. Equal set
Hint
Equal sets in probability mean Element in both the sets are equal.

Question 36.
Three unbiased coins are tossed. The probability of obtaining at least one head is
a. 1/8
b. 2/8
c. 6/8
d. 7/8
Answer:
d. 7/8
Hint
Probability of an event = m/m+n
Where m= number of outcomes in favor of event
m+n=total number of equally likely mutually exclusive and exhaustive events finding m
total number of coins = 2 × 2 × 2=8
there is only one chance when no head appears which is tail,tail, tail
thus the number of out comes in favor of event = 8-1 =7
m=7
probability = 7/8

Question 37.
A bag contains 4 white, 5 red, and ~ blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is
a. 1/15
b. 2/91
c. 3/107
d. 15/107
Answer:
b. 2/91
Hint
Probability of all three balls to be red = 5 × 4 × 3/15 × 14 × 13= 2/91 a

Question 38.
An event that corresponds to a single possible outcome of an experiment is caned:
a. Elementary event
b. Compound event
c. Dependent event
d. Exhaustive event
Answer:
a. Elementary event
Hint
In probability theory, an elementary event is an event that contains only a single outcome in the sample space.

Question 39.
if a card is drawn at random from a pack of 52 cards, the chance that it will be a king of hearts is:
a. 1/13
b. 4/13
c. 1/52
d. 4/26
Answer:
c. 1/52
Hint
Only one king of hearts Probability = 1/52

Question 40.
The total number of all possible outcomes of a random experiment constitutes:
a. Equally likely events
b. Exhaustive events
c. Mutually inclusive events
d. None of the above
Answer:
b. Exhaustive events
Hint
One or more events are said to be exhaustive if all the possible elementary events under the experiment are covered by the events considered together. In other words, the events are said to be exhaustive when they are such that at least one of the events compulsorily occurs.

Question 41.
What is the probability of getting a sum of 9 from two throws of a dice?
a. 1/9
b. 2/9
c. 1/36
d. 1/3
Answer:
a. 1/9
Hint
Probability = no. of favourable events/total no. of events = 4/36 = 1/9
No. of favourable events = (3,6), (4,5),(5,4),(6,3)

Question 42.
Mutually exclusive events mean –
a. No events can be expected to occur in preference to any other event in the same experiment
b. Events that can be decomposed further into elementary events
c. The occurrence of one event prevents the occurrence of another event in the same experiment
d. Events that are independent of one another.
Answer:
c. The occurrence of one event prevents the occurrence of another event in the same experiment
Hint
Mutually Exclusive means you can’t get both events at the same time. It is either one or the other, but not both

Question 43.
A bag contains 6 black and 8 white balls. One ball is drawn at random. The probability that the ball drawn is white will be
a. 1/14
b. 1/7
c. 3/7
d. 4/7
Answer:
d. 4/7
Hint
Total balls = 14 Favourable evnts = 8 P = 8/14 = 4/7

Question 44.
When 2 coins are tossed, then what is the probability of getting one head?
a. 1/2
b. 1/4
c. 1
d. 3/4
Answer:
a. 1/2
Hint
No. of possible outcome = 2
Total no. of outcome =4 (HH,HT,TH,TT)
P = 2/4 =1/2

Question 45.
Which set is it in which no elements are present?
a. Sub set
b. Null set
c. Disjoint
d. Equal set
Answer:
b. Null set
Hint
The null set, also called the empty set, is the set that does not contain anything.

Question 46.
If a dice is thrown once, what is the probability of getting an even number?
a. 1/2
b. 1
c. 2/6
d. None of the above
Answer:
a. 1/2
Hint
Total no. of outcome = 6
No. of even favourable outcome = 3 (2,4,6)
P = 3/6 =1/2

Question 47.
What is the probability of getting a queen from a pack of cards?
a. 1/52
b. 4/52
C. 2/13
d. 13/52
Answer:
b. 4/52
Hint
Total no. of outcome = 52
No. of favorable outcome of getting queen = 4
P = 4/52

Question 48.
What is the probability of getting king of heart from the pack of cards?
a. 1/52
b. 13/52
c. 2/52
d. 2/13
Answer:
a. 1/52
Hint
Total no. of outcome = 52
No. of favorable outcome (rings of the heart) = 1
P = 1/52

Question 49.
Find the probability that a leap year has 53 Sundays.
a. 2/7
b. 54
C. 1/9
d. 53
Answer:
a. 2/7
Hint
There are 366 days in leap year out of Which there are 52 Sundays and 2 days.
2 days may be (SU,M)(M,T)(T,W)(W,TH)(TH,F)(F,S)(S,SU)
Thus total outcome =7
No. of favourable outcome = 2 (SU,M) and (S,SU)
P= 2/7

Question 50.
Given A = (1,2,3,4.5) & B = (4,3,1,5,6). find (B-A)
a. 6
b. 2
c. 1
d. 3
Answer:
a. 6
Hint
B-A means elements of B not included in A = 6

Question 51.
If set A = {1 ,2,3,4}, 8 = {1 ,2,} Then, which one is correct?
a. Set A is a subset of set 8
b. Set 8 is a subset of set A
C. 80th (a).& (b)
d. None of the above
Answer:
b. Set 8 is a subset of set A
Hint
Set B is a subset of set A because both elements 1,2 are present in set A.

Question 52.
In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?
a. 10/21
b. 7/21
c. 8/21
d. 15/21
Answer:
b. 7/21
Hint
Total no. of outcome = 8+7+6=21 No. of favorable outcome =7 P= 7/21

Question 53.
A town has a total population of 50,000. Out of it, 28,000 read the newspaper X and 23,000 read Y while 4,000 read both the papers. The number of persons not reading X and Y both are:
a. 3,500
b. 2,500
c. 3,000
d. 2,000
Answer:
c. 3,000
Hint
n(AUB) = n(A)+n(B)-n(ADB)
= 28000+23000-4000 = 47000 n(ADB) =50000-47000 = 3000

Question 54.
If all the elements of set A belongs to set B and all the elements of set B belong to set A they can be referred to as:
a. Equal sets
b. Supersets
c. Disjoint sets
d. Subsets
Answer:
a. Equal sets
Hint
Two sets are equal if they have exactly the same elements.

Question 55.
Two coins are tossed, find the probability that two heads are obtained:
a. 0.75
b. 0.25
c. 1
d. 0.50
Answer:
d. 0.50
Hint
Total outcomes = 4
No. of favouable outcome =2
P = 2/4 =1/2= 5

CS Foundation Business Economics Notes