Module 5

Question 1

What do you need to calculate probability?

Answer 1

The number of trials (n) and the number of outcomes per trial (k).

Question 2

What is the formula for multistep probability?

Answer 2

Multiply the number of outcomes per trial for all trials. Example: For 3 coins, (2)(2)(2)=8.

Question 3

What is 5!?

Answer 3

5!=5×4×3×2×1=120.

Question 4

What is a discrete uniform distribution?

Answer 4

A distribution where all outcomes are equally likely, like rolling a die.

Question 5

How do you validate a probability distribution?

Answer 5

• Probabilities must be between 0 and 1.
• The sum of probabilities must equal 1.

Question 6

What is the probability of rolling two 6’s with two dice?

Answer 6

Total outcomes = 6×6=36.
Probability of two 6’s = 1/36
.

Question 7

How many possible outcomes are there when rolling 5 dice?

Answer 7

(6)(6)(6)(6)(6) = 6^5 =7776

Question 8

What are the conditions for a discrete probability distribution?

Answer 8

0<P(X=x)<1 for all outcomes.
∑P(X=x)=1.

Question 9

What is a random variable?

Answer 9

A variable that takes on numerical values based on the outcomes of a random event

Question 10

What is a probability mass function (PMF)?

Answer 10

A function that assigns probabilities to each value of a discrete random variable.

Question 11

What is the relationship between trials and outcomes in probability?

Answer 11

Multiply the number of outcomes for each trial to find total outcomes.

Question 12

What are the four types of probability distributions?

Answer 12

Discrete distributions
Binomial
Poisson

Continuous distributions
Normal
Exponential

Question 13

What are some examples of a binomial distribution?

Answer 13

A customer defaults or does not default on a loan.

A consumer reacts positively or negatively to a social media campaign.

A drug is either effective or ineffective.

Question 14

What are the components of a binomial distribution?

Answer 14

n: Number of trials.

x: Number of successes.

P: Probability of success.

1−P: Probability of failure.

Question 15

What is the formula for a binomial probability?

Answer 15

P(X=x)= (n x) P^x (1– P)^n-x

(n x) = number of ways to choose x successes out of n trials
P^z = probability of x successes
(1 – P)^n-x = probability of n–x failures

Question 16

What are the properties of a binomial experiment?

Answer 16

Fixed number of trials (n).

Each trial is independent.

Only two possible outcomes (success/failure).

The probability of success (P) is the same for each trial.

Question 17

What is the difference between discrete and continuous probability distributions?

Answer 17

Discrete: Outcomes are countable (e.g., Binomial, Poisson).

Continuous: Outcomes are uncountable and measured over a range (e.g., Normal, Exponential).

Question 18

Binomial Distribution

Answer 18

Discrete probability distribution for a fixed number of independent trials, each with two possible outcomes.

Question 19

Poisson Distribution

Answer 19

Describes the number of successes in a fixed time or space.
Examples:

Spam emails in a month.
Cars in line at McDonald’s in an hour.

Question 20

Expected Value

Answer 20

The weighted average of all possible values for a random variable.

Question 21

Variance

Answer 21

Measures spread around the mean.

Question 22

Risk Preferences

Answer 22

Risk-Averse: Avoid risk without positive gain.

Risk-Neutral: Focus on expected gain only.

Risk-Loving: Willing to accept risk, even with negative gain.

Question 23

Definition of the Poisson Distribution

Answer 23

Used for counting the number of occurrences (successes) of an event over a specific interval of time or space.

A Poisson process satisfies these conditions:

The number of successes in an interval can be any integer from 0 to infinity.

Successes in non-overlapping intervals are independent.

The probability of success is proportional to the size of the interval.

Question 24

Excel Formula for Poisson Distribution

Answer 24

Syntax: POISSON.DIST(x, mean, cumulative)

x: The number of occurrences (successes).

mean: The expected value (average rate of occurrence).

Cumulative:
-TRUE: Returns cumulative probability P(X ≤ x).
-FALSE: Returns probability mass P(X = x).

Question 25

Example: Mean = 15, Find P(X=10)

Answer 25

Use the formula: POISSON.DIST(10, 15, FALSE).

Question 26

Key Properties of Poisson Random Variable

Answer 26

Discrete probability distribution.
Expected value (mean): λ
Variance: λ.

Question 27

Binomial vs. Poisson

Answer 27

Binomial Probability: P(6≤X≤11)=P(X≤11)−P(X≤5).
Used for finite trials with success probability (e.g., graduate degree example).

Poisson:
Used for events occurring over time or space (e.g., foreclosures).