WorksheetsProbability Theory and Combinatorics - Key Facts and Definitions
Total questions: 87
Worksheet time: 44mins
An event that always occurs in every trial is called ________.
Certain (sure) event.
Impossible event.
Random event.
Unlikely event.
The classical definition of probability was given by ________.
Pierre-Simon Laplace.
Isaac Newton.
Blaise Pascal.
Carl Friedrich Gauss.
The union of events A and B means ________.
A or B or both occur.
Only A occurs.
Only B occurs.
Neither A nor B occurs.
The intersection of events A and B means ________.
Both A and B occur simultaneously.
Only A occurs, not B.
Only B occurs, not A.
Either A or B occurs, but not both.
The difference of events A and B (A \ B) means ________.
A occurs but B does not.
Both A and B occur.
B occurs but A does not.
Neither A nor B occurs.
Combinatorics is ________.
The branch of mathematics that studies counting, arrangement, and selection of elements.
A type of geometric transformation.
A method for solving differential equations.
A branch of mathematics focused on calculus.
The main problems of combinatorics are related to ________.
Permutations, placements, and combinations.
Algebraic equations and inequalities.
Geometric constructions and proofs.
Calculus and differential equations.
An example of a combinatorial problem is ________.
Counting the number of possible passwords.
Solving a quadratic equation.
Drawing a straight line.
Measuring temperature.
For mutually exclusive events A and B, the probability of their union equals ________.
P(A ∪ B) = P(A) + P(B).
P(A ∪ B) = P(A) × P(B).
P(A ∪ B) = P(A) - P(B).
P(A ∪ B) = P(A) / P(B).
Events that exclude each other’s occurrence are ________.
Mutually exclusive events.
Independent events.
Complementary events.
Simultaneous events.
For independent events A and B, the probability of their joint occurrence equals ________.
P(A ∩ B) = P(A) × P(B).
P(A ∩ B) = P(A) + P(B).
P(A ∩ B) = P(A) - P(B).
P(A ∩ B) = P(A) / P(B).
Conditional probability of A given B is ________.
P(A | B) = P(A ∩ B) / P(B).
P(A | B) = P(A) / P(B).
P(A | B) = P(B) / P(A).
P(A | B) = P(A ∪ B) / P(B).
Probability of a certain (sure) event is ________.
1.
0.
0.5
-1
Probability of drawing a red ball from urn (9 red, 6 yellow, 5 green) is ________.
9/20.
6/20.
5/20.
1/3.
Out of 1000 TVs, 5 defective. The probability of working TV is ________.
0.995.
0.950.
0.500.
0.900.
Probability that a team enters group B (4 groups) is ________.
1/4.
1/2.
1/3.
1/5.
8 tourists choose 6 to go shopping. The probability tourist D goes is ________.
6/8 = 3/4.
1/8.
2/8 = 1/4.
5/8.
In a class of 16, what is the probability that 2 friends are in the same group of 4?
1/5
1/4
1/6
1/3
What is the probability of the opposite (complementary) event?
P(Ā) = 1 – P(A)
P(Ā) = P(A) + 1
P(Ā) = 1 / P(A)
P(Ā) = P(A) – 1
If a clock hand stops randomly between 6 and 9, what is the probability?
3/12 = 1/4
1/2
1/3
1/6
What is the probability of mutually exclusive events?
Sum of their individual probabilities.
Product of their individual probabilities.
Difference of their individual probabilities.
Zero.
What is the sum of probabilities of all elementary outcomes?
1
0
0.5
2
Give an example of mutually exclusive events.
Getting heads or tails in one coin toss.
Getting heads and tails in one coin toss.
Getting a number greater than 2 and less than 2 on a die roll.
Getting red and black card in a single card draw.
Give an example of equally likely events.
Rolling any number on a fair die.
Getting a head and getting a tail when tossing a biased coin.
Drawing a red card from a deck of only black cards.
Picking a king from a deck of cards with no kings.
In geometric probability, the measure of the region represents what?
Length, area, or volume.
Probability of an event occurring.
Number of possible outcomes.
The sum of all probabilities.
What is the true statement for geometric probability?
P(A) = (measure of favorable region) / (measure of total region).
P(A) = (measure of total region) / (measure of favorable region).
P(A) = (measure of favorable region) × (measure of total region).
P(A) = (measure of favorable region) + (measure of total region).
Combinatorics is applied in which fields?
Math, computer science, statistics, biology, cryptography.
Only in art and music.
Exclusively in sports and entertainment.
Mainly in cooking and fashion.
For unrepeatable combinations (dials), how is each element used?
Each element used once; order doesn’t matter.
Each element used multiple times; order matters.
Each element used once; order matters.
Each element used multiple times; order doesn’t matter.
For mutually exclusive events, what is true about their occurrence?
They can’t occur together; P(A ∩ B) = 0.
They always occur together; P(A ∩ B) = 1.
They are independent; P(A ∩ B) = P(A)P(B).
They can occur together with some probability.
For dependent events, what is true?
One affects the probability of the other.
They are always mutually exclusive.
They have equal probabilities.
They cannot occur together.
For compatible events, what is true about their occurrence?
They can occur together; P(A ∩ B) ≠ 0.
They cannot occur together; P(A ∩ B) = 0.
They are always mutually exclusive.
Their probabilities always add up to 1.
What is true about mutually exclusive events?
Cannot occur at the same time.
Always occur together.
Are independent of each other.
Have a probability sum greater than 1.
Which of the following are true probability statements?
0 ≤ P(A) ≤ 1
P(Ω) = 1
P(∅) = 0
For equally likely outcomes, what is true about each outcome?
Each outcome has same probability.
Each outcome has a different probability.
Each outcome is impossible.
Each outcome is certain.
What are the properties of geometric definition?
Non-negativity, normalization, additivity.
Commutativity, distributivity, associativity.
Transitivity, reflexivity, symmetry.
Continuity, differentiability, integrability.
What is a random event?
May or may not occur under same conditions.
Always occurs under same conditions.
Never occurs under any conditions.
Occurs only once in a lifetime.
Fill in the blank: Elementary events cannot be decomposed; they are ______ and equally likely.
distinct
random
dependent
infinite
Fill in the blank: Repeated combinations are used when ______ can repeat, order doesn’t matter.
elements
numbers
positions
choices
Fill in the blank: An example of repeated combinations is ______ with repeated ingredients.
forming ice cream flavors
arranging books on a shelf
drawing cards without replacement
assigning seats in a theater
Fill in the blank: The addition theorem is used to find probability that at least ______ occurs.
one event
two events
no event
all events
Fill in the blank: For independent events, P(A ∩ B) = P(A) × P(B); one doesn’t ______ the other.
influence
exclude
replace
follow
Fill in the blank: The formula of total probability is used when an event can occur in several ______ ways.
mutually exclusive
independent
dependent
identical
Fill in the blank: Dependent events are when the occurrence of one changes the probability of ______.
another
itself
the past
the outcome
Fill in the blank: General probability statements: Probability is between 0 and 1; sum of all outcomes = ______.
1
0
2
-1
Which of the following is a classical probability example?
Coin toss
Dice roll
Card draw
All of the above
Fill in the blank: Events are dependent when one ______ the likelihood of another.
affects
ignores
decreases
multiplies
Fill in the blank: Features of a random event include uncertainty, repeatability, and ______ probability.
measurable
infinite
subjective
unknown
Which of the following is an example of mutually exclusive events?
Rolling 3 or 5 on one die
Drawing king or queen from one card
Both A and B
None of the above
Fill in the blank: Combinatorics is the study of ______, arranging, and selecting.
counting
measuring
drawing
mixing
Fill in the blank: The main problems of combinatorics are permutations, placements, and ______.
combinations
equations
derivatives
integrals
Fill in the blank: Unrepeatable combinations mean each element is used once; ______ doesn’t matter.
order
repetition
position
value
Fill in the blank: Repeated combinations mean repetition allowed; ______ doesn’t matter.
order
number
sum
value
Combinatorics is applied in which of the following fields?
Mathematics
Computer science
Biology
All of the above
Fill in the blank: The formula for combinations without repetition is C(n, k) = n! / [k! (n–k)!].
C(n, k) = n! / [k! (n–k)!]
C(n, k) = n! / (n–k)!
C(n, k) = k! / [n! (n–k)!]
C(n, k) = n! / k!
Example of combinations with repetition — ________.
Choosing ice cream flavors with repetition.
Arranging books on a shelf without repetition.
Selecting a president and vice president from a group.
Assigning unique ID numbers to students.
Diagonals in a convex 12-sided polygon — ________.
54.
60.
66.
72.
Two balls into two boxes (with repetition) — ________.
4 ways.
2 ways.
3 ways.
5 ways.
Unrepeatable combinations statements — ________.
Order not important; no repetition.
Order important; repetition allowed.
Order not important; repetition allowed.
Order important; no repetition.
Combinatorics in applied sciences — ________.
Programming, data analysis, networks, genetics.
Painting, sculpture, music, dance.
Cooking, gardening, pottery, knitting.
Poetry, drama, fiction, essays.
Repeated combinations conditions — ________.
Repetition allowed; order irrelevant.
Repetition not allowed; order relevant.
Repetition not allowed; order irrelevant.
Repetition allowed; order relevant.
Operations related to combinatorics — ________.
Arranging, selecting, distributing.
Measuring, weighing, mixing.
Drawing, painting, sculpting.
Cooking, baking, frying.
Characteristics of combinatorial problems — ________.
Finite outcomes; counting arrangements/selections.
Infinite outcomes; continuous variables.
Random sampling; probability distributions.
Optimization of continuous functions.
Main types of combinatorial tasks — ________.
Permutations, arrangements, combinations.
Addition, subtraction, multiplication.
Graphs, trees, networks.
Equations, inequalities, functions.
Difference between repeated and unrepeatable — ________.
Repetition allowed vs not allowed.
Both allow repetition.
Neither allows repetition.
Unrepeatable allows more repetition.
Combinatorial applications — ________.
Passwords, scheduling, seat plans, genetic codes.
Cooking recipes, musical notes, painting colors, dance moves.
Weather patterns, ocean currents, mountain ranges, river flows.
Historical events, famous speeches, ancient ruins, old manuscripts.
Unrepeatable combinations represent — ________.
Selections without repetition.
Selections with repetition.
Arrangements with repetition.
Arrangements without repetition.
Examples of repeated combinations — ________.
Pizza toppings with same ingredients.
Arranging books in a row.
Assigning seats to students.
Selecting a team captain.
Conditional probability of A given B — ________.
P(A|B) = P(A ∩ B)/P(B).
P(A|B) = P(A) + P(B).
P(A|B) = P(B)/P(A ∩ B).
P(A|B) = P(A) × P(B).
Multiplication theorem — ________.
For joint occurrence of events.
For mutually exclusive events.
For independent events only.
For single event probability.
Addition theorem — ________.
For probability of at least one event occurring.
For probability of all events occurring together.
For probability of mutually exclusive events only.
For probability of independent events only.
Events A and B incompatible if — ________.
Cannot occur together.
Are mutually exclusive.
Always occur together.
Are independent events.
Probability of incompatible A and B — ________.
P(A ∪ B) = P(A) + P(B).
P(A ∩ B) = P(A) + P(B).
P(A ∪ B) = P(A) × P(B).
P(A ∩ B) = P(A) × P(B).
Probability of compatible events’ union — ________.
P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
P(A ∪ B) = P(A) × P(B).
P(A ∪ B) = P(A) + P(B) + P(A ∩ B).
P(A ∪ B) = P(A) – P(B) + P(A ∩ B).
Symbol for intersection probability — ________.
P(A ∩ B).
P(A ∪ B).
P(A) + P(B).
P(A | B).
H₁, H₂, ..., Hₙ form complete group if — ________.
Mutually exclusive and exhaustive.
Independent and exhaustive.
Collectively exhaustive and dependent.
Mutually independent and exclusive.
Total probability formula used when — ________.
Event can occur via several exclusive ways.
Event is certain to happen.
Events are independent.
Events are mutually inclusive.
H1, H2, ..., Hn are called — ________.
Complete group (partition) of events.
Independent events.
Mutually exclusive events.
Random variables.
True for mutually exclusive events — ________.
Cannot happen together; P(A ∩ B) = 0.
Can always happen together; P(A ∩ B) = 1.
Are independent; P(A ∩ B) = P(A)P(B).
Must be exhaustive events.
Multiplication theorem applies to — ________.
Joint occurrence of events.
Mutually exclusive events.
Independent events only.
Impossible events.
For complete group of events — ________.
Sum of probabilities = 1.
Sum of probabilities = 0.
Sum of probabilities = 2.
Sum of probabilities = 0.5.
To use total probability — ________.
Complete group + conditional probabilities required.
Only independent events required.
Only mutually exclusive events required.
No conditions required.
Conditional probability examples — ________.
Disease test, rain given clouds.
Multiplying two numbers, adding fractions.
Drawing a straight line, measuring angles.
Counting apples, naming colors.
Total probability applies when — ________.
Event depends on several possible causes.
Event is certain to occur.
Event is independent of any cause.
Event cannot be measured.
Properties of conditional probability — ________.
0 ≤ P(A|B) ≤ 1; P(A|B) = P(A∩B)/P(B).
P(A|B) = P(A) + P(B).
P(A|B) = P(A) - P(B).
P(A|B) = P(A∪B)/P(B).
Addition theorem forms — ________.
For incompatible: P(A ∪ B)=P(A)+P(B); compatible: minus P(A∩B).
For all events: P(A ∩ B)=P(A)+P(B).
For independent: P(A ∪ B)=P(A)×P(B).
For exclusive: P(A ∪ B)=P(A)−P(B).
Dependent events — ________.
P(A ∩ B)=P(A)×P(B|A); one affects another.
P(A ∩ B)=P(A)+P(B); events are unrelated.
P(A ∩ B)=P(A)×P(B); events are independent.
P(A ∩ B)=P(A)-P(B); events are mutually exclusive.
Elements needed for total probability — ________.
Complete event group H1,...,Hn and P(A|Hi).
Only the probability of A.
Just one event from the group.
The sum of all probabilities without conditions.
