Probability theory

Question 1

What is probability?

Answer 1

Probability is the likelihood that a particular event will occur.

Question 2

What is the mathematical formula for probability?

Answer 2

P(A)=n(A)/n(s)
​
n(A) = number of favorable outcomes
n(S) = total number of outcomes

Question 3

What are the two types of probability?

Answer 3

Subjective Probability: Based on intuition, opinion, or experience.
Objective Probability: Based on empirical data or statistical analysis.

Question 4

What is an experiment in probability?

Answer 4

A planned operation carried out to observe outcomes.

Question 5

What is a random experiment?

Answer 5

An experiment where the outcome is uncertain.

Question 6

What is a sample space?

Answer 6

The set of all possible outcomes of an experiment.

Question 7

Give an example of a sample space for flipping a coin.

Answer 7

S={H,T}
where H = Heads, T = Tails.

Question 8

What is an event in probability?

Answer 8

A subset of the sample space.

Question 9

What is the complement of an event?

Answer 9

The set of all outcomes not in event
𝐴
A.

Question 10

What is the formula for the complement of an event?

Answer 10

P(A)+P(A′)=1

Question 11

If an event A has a probability of 0.65, what is the probability of A′?

Answer 11

P(A′)=1−P(A)=1−0.65=0.35

Question 12

What are the three ways to determine probability?

Answer 12

A Priori Probability: When outcomes are known in advance (e.g., rolling a die).
Empirical Probability: Based on observed data.
Mathematical Probability: Using probability distributions

Question 13

What is the range of probability values?

Answer 13

Probability values range between 0 and
0≤P(A)≤1

Question 14

Differentiate between these type of events. (Certain, impossible, simple, compound, mutually exclusive, independent, dependent)

Answer 14

Certain: An event that will definitely happen, P(A)=1.
Impossible: An event that cannot happen, P(A)=0.
Simple: An event with only one possible outcome.
Compound: An event that includes multiple possible outcomes.
Mutually Exclusive: Events that cannot happen at the same time.
Independent: Events where the outcome of one does not affect the other.
Dependent: Events where the outcome of one affects the probability of the other.

Question 15

Write the following probability rules: Addition for mutually exclusive, addition for non-mutually exclusive, multiplication rule for independent events, multiplication rule for dependent.

Answer 15

Addition for mutually exclusive: P(A∪B)=P(A)+P(B)
Addition for non-mutually exclusive: P(A∪B)=P(A)+P(B)−P(A∩B)
Multiplication rule for independent events: P(A∩B)=P(A)×P(B)
Multiplication rule for dependent: P(A∩B)=P(A∣B)×P(B)

Question 16

What is conditional probability and its formula?

Answer 16

The probability of event A occurring given that event B has already occurred.

P(A∣B)= P(B)/P(A∩B)

Question 17

What does it mean if
P(A∣B)=0.4?

Answer 17

It means that given that B has already occurred, there is a 40% chance that A will also occur.

Question 18

Differentiate between joint and marginal probability.

Answer 18

Joint: The probability of two events happening together in one trial.

Marginal: The probability of a single event occurring without considering another event.

Question 19

What is the probability that an event will happen at least once in multiple trials?

Answer 19

Use the complement rule:
P(Atleast1success)=1−P(Nosuccessinanytrial)

Question 20

Explain the difference between mutually exclusive and independent events.

Answer 20

Mutually exclusive events cannot happen at the same time (e.g., rolling a 3 and a 4 on a die in one roll).
Independent events do not affect each other (e.g., flipping a coin and rolling a die).