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Question 1

What is Bayes’ Theorem primarily used for?

Answer 1

To answer questions about the role of one event’s occurrence in relation to another event

Question 2

Define independent events in probability.

Answer 2

If P(B | A) = P(B), then A and B are independent events

Question 3

What is the formula for conditional probability P(B | A)?

Answer 3

P(B | A) = P(A ∩ B) / P(A)

Question 4

What does it mean if an event has a probability of 1?

Answer 4

The event is called a certain or sure event

Question 5

If the probability of an event is zero, what is it called?

Answer 5

An impossible event

Question 6

What is the range of probability values for any event A in sample space S?

Answer 6

0 ≤ P(A) ≤ 1

Question 7

What is the formula for the union of two events A and B?

Answer 7

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Question 8

Fill in the blank: If a trial results in n exhaustive mutually exclusive and equally likely events, the probability of event A is _______.

Answer 8

m/n

Question 9

What is the Total Probability Theorem used for?

Answer 9

To calculate the probability of an event when conditional probabilities are known

Question 10

What is meant by a ‘prior probability’?

Answer 10

P(A) is called ‘A prior Probability’ because it exists before gaining any information from the experiment

Question 11

Define ‘posterior probability’.

Answer 11

P(Ai | B) is called ‘Posterior probability’ determined after knowing the results of the experiment

Question 12

What is a random variable?

Answer 12

A real number x connected with an outcome of a random experiment E

Question 13

What characterizes a discrete random variable?

Answer 13

It takes at most a countable number of values

Question 14

What is the probability mass function (pmf)?

Answer 14

The probability function associated with discrete random variables

Question 15

What is a continuous random variable?

Answer 15

A random variable that can take all possible values between certain limits

Question 16

What is the probability density function (pdf)?

Answer 16

The probability function associated with continuous random variables

Question 17

Fill in the blank: The variance of a random variable X is defined as _______.

Answer 17

E[(X - µ)²]

Question 18

What does ‘sensitivity’ refer to in the context of diagnostic tests?

Answer 18

The probability of a positive test result given the presence of the disease

Question 19

What is the probability that a randomly chosen individual from a population has at least one mutation if 40% have a wing mutation, 20% have an eye mutation, and 12% have both?

Answer 19

P(at least one mutation) = P(wing) + P(eye) - P(both)

Question 20

True or False: If two events A and B are mutually exclusive, they can occur at the same time.

Answer 20

False

Question 21

What is the formula for the intersection of two events A and B?

Answer 21

P(A ∩ B) = P(A) P(B | A) or P(B) P(A | B)

Question 22

What is the expected value (mean) of a random variable X?

Answer 22

E(X) = Σ [x * P(x)] for discrete; ∫ x * f(x) dx for continuous

Question 23

What does the complement of an event A represent?

Answer 23

The event that A does not occur, denoted as A′

Question 24

What is the definition of an event in probability?

Answer 24

An event is a subset A of the sample space S, representing a set of possible outcomes.

Question 25

What are the two types of events based on their occurrence?

Answer 25

An event can either occur or not occur.

Question 26

What is a simple or elementary event?

Answer 26

An event consisting of a single point of S.

Question 27

What is the sure or certain event in probability?

Answer 27

The sample space S itself.

Question 28

What is the impossible event in probability?

Answer 28

The empty set ∅.

Question 29

What does A ∪ B represent in probability?

Answer 29

A ∪ B is the event 'either A or B or both,' called the union of A and B.

Question 30

What does A ∩ B represent in probability?

Answer 30

A ∩ B is the event 'both A and B,' called the intersection of A and B.

Question 31

What is the complement of an event A?

Answer 31

A′ is the event 'not A', and A′ = S - A.

Question 32

What does A - B represent?

Answer 32

A - B = A ∩ B′ is the event 'A but not B.'

Question 33

What are mutually exclusive events?

Answer 33

Events are mutually exclusive if A ∩ B = ∅, meaning they cannot both occur.

Question 34

What is a random experiment?

Answer 34

An experiment where results can vary from one performance to another under nearly identical conditions.

Question 35

What is a sample space?

Answer 35

A set S that consists of all possible outcomes of a random experiment.

Question 36

What is the sample space for flipping a coin?

Answer 36

{H, T}.

Question 37

What is the sample space for rolling a die?

Answer 37

{1, 2, 3, 4, 5, 6}.

Question 38

What is the sample space for playing a football match?

Answer 38

{Win, Draw, Lose}.

Question 39

Fill in the blank: Each performance in a random experiment is called a _______.

Answer 39

[trial].

Question 40

What is an outcome in the context of a random experiment?

Answer 40

The result of a trial in a random experiment, also known as an elementary event or sample point.

Question 41

What are equally likely events?

Answer 41

Outcomes that have no reason to expect one in preference to the other.

Question 42

List examples of random experiments.

Answer 42

* Flipping a coin * Rolling a die * Sexual intercourse for reproduction * Gambling * Surgical procedure

Question 43

What is the focus of probability theory?

Answer 43

The analysis of phenomena that take place in indeterministic or random circumstances.

Question 44

What does the theory of probability provide?

Answer 44

Mathematical models for real-world phenomena involving randomness.