Module 4

Question 1

What are exhaustive events in probability?

Answer 1

Exhaustive events include all possible outcomes of an experiment.
They cover the entire sample space.

Question 2

What are mutually exclusive events in probability?

Answer 2

Mutually exclusive events do not share any common outcomes.
The occurrence of one event precludes the occurrence of the other.

Question 3

Are the following events mutually exclusive?
Grades of A, B, C

Answer 3

Yes, because a student cannot receive both an A and a B simultaneously.
However, they are not exhaustive, as other grades (D, F) are possible.

Question 4

What is the union (A ∪ B) of two events?

Answer 4

The union of two events includes all outcomes in either A or B, without double-counting.
Example: A ∪ B = {gold, silver, bronze, no medal}.

Question 5

What is the intersection (A ∩ B) of two events?

Answer 5

The intersection of two events includes only the common outcomes in both events.
Example: A ∩ B = {silver, bronze}.

Question 6

What is the complement (Bc) of an event?

Answer 6

The complement of an event includes all outcomes not in the event.
Example: Bc = {gold}, if B = {silver, bronze, no medal}.

Question 7

How do you calculate P(A) for a snowboarder’s medal probabilities?

Answer 7

P(A) = P({gold}) + P({silver}) + P({bronze}) = 0.10 + 0.15 + 0.20 = 0.45.

Question 8

What is empirical probability?

Answer 8

Empirical probability is based on observing data or the relative frequency of an event occurring.
It’s calculated by repeating an experiment many times.

Question 9

How would you calculate the probability that an individual is between 50 and 60 years old, based on a frequency distribution?

Answer 9

Use the relative frequency formula:
P(50 ≤ age < 60) = (Number of individuals aged 50–60) / (Total number of observations).

Question 10

Given sample space S = {gold, silver, bronze, no medal}, how do you calculate P(B ∪ C)?

Answer 10

P(B ∪ C) = P({silver}) + P({bronze}) + P({no medal}) = 0.15 + 0.20 + 0.55 = 0.90.

Question 11

What is the rule of addition for calculating the probability of at least one event happening?

Answer 11

The probability of event A or event B happening is:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.75 + 0.55 - 0.40 = 0.90.

Question 12

How do you calculate the probability that no event occurs?

Answer 12

The probability that neither A nor B happens is the complement:
P(not A ∩ not B) = 1 - P(A ∪ B).
1 - 0.90 = 0.10.

Question 13

What is conditional probability?

Answer 13

The probability of event A occurring given that event B has already occurred is:
P(A | B) = P(A ∩ B) / P(B).

Question 14

How do you determine if two events are independent?

Answer 14

Two events A and B are independent if:
P(A ∩ B) = P(A) * P(B).

Example: If the probability of your desktop crashing is 0.02, and the probability of your laptop crashing is 0.06, and the probability of both crashing is 0.0012, check if they are independent:
P(A) * P(B) = 0.02 * 0.06 = 0.0012 → They are independent.

Question 15

How do you calculate the probability of both events A and B occurring?

Answer 15

The probability of both events occurring is:
P(A ∩ B) = P(A) * P(B) for independent events.

If they are not independent, use the rule of addition:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B).

Question 16

What are sample spaces and events?

Answer 16

Sample Space (S): All possible outcomes of an experiment.
Event: A subset of the sample space (e.g., A, B).

Question 17

What is the difference between exhaustive and mutually exclusive events?

Answer 17

Exhaustive Events: Include all possible outcomes in the sample space.
Mutually Exclusive Events: Cannot occur simultaneously (no common outcomes).

Question 18

What is the addition rule?

Answer 18

The probability of event A or event B occurring:
P(A∪B)=P(A)+P(B)−P(A∩B)

Example:
P(A) = 0.75, P(B) = 0.55, P(A ∩ B) =
P(A∪B)=0.75+0.55−0.40=0.90

Question 19

What is the complement rule?

Answer 19

The probability of an event not occurring:

P(notA)=1−P(A)
Example:
If P(A) = 0.90, then P(not A) = 1 - 0.90 = 0.10.

Question 20

What is conditional probability?

Answer 20

The probability of event A given that event B has occurred:

P(A∣B)= P(B)/P(A∩B)

Example:
P(A ∩ B) = 0.16, P(B) = 0.25

P(A∣B)= 0.25/0.16 =0.64
​

Question 21

Joint probability

Answer 21

Joint Probability (P(A ∩ B)): The probability that both events A and B occur together.

Question 22

Union probability

Answer 22

Union Probability (P(A ∪ B)): The probability that either event A or event B (or both) occurs.

P(A∪B)=P(A)+P(B)−P(A∩B)

Question 23

What is a contingency table, and how is it used in statistics?

Answer 23

A contingency table displays the frequency distribution of two or more categorical variables. It helps calculate joint, marginal, and conditional probabilities by organizing data into rows and columns.

Question 24

What is the difference between joint, marginal, and conditional probability?

Answer 24

Joint Probability: Probability of two events occurring together (e.g., both being male and promoted).

Marginal Probability: Probability of a single event (e.g., probability of being promoted).

Conditional Probability: Probability of one event given that another has already occurred (e.g., probability of being promoted given gender).

Question 25

How do you calculate the probability of multiple independent events occurring?

Answer 25

Multiply the number of possible outcomes for each event.

Example: Rolling 3 coins:
2×2×2=8 outcomes.

Question 26

What is the formula for calculating factorials?

Answer 26

n!=n×(n−1)×(n−2)×⋯×1

Example: 4!=4×3×2×1=24

Question 27

What is the difference between combinations and permutations?

Answer 27

Combinations: Order doesn’t matter

Permutations: Order matters

Question 28

How do you calculate conditional probability?

Answer 28

What is the probability that a customer prefers chocolate frozen yogurt, given they are at least 25 years old?

Formula: P(C∣T)=P(C∩T)/P(T)

Result: P(C∣T) = 15/75 = 0.20

Question 29

How do you calculate the probability of an event?

Answer 29

P(A) = number of favorable outcomes/total number of outcomes

Question 30

What are the counting rules for calculating possible outcomes?

Answer 30

Multistep: Multiply the number of outcomes for each step.

Factorial: Use n! to calculate the number of ways to arrange objects.

Combination: Used when order doesn’t matter, C(n,k).

Permutation: Used when order matters, P(n,k).

Question 31

How do you calculate the probability of multiple events?

Answer 31

Multiply the individual probabilities for independent events.