# Probability basics imported Flashcards

1
Q

What are probability distributions?

A

Probability distributions are mathematical representations that describing the likelihood of each outcomes of a random experiment or process.

2
Q

How are probability distributions expressed?

A

Probability distributions are expressed using mathematical functions or equations.

3
Q

What is an example of a probability distribution?

A

An example is the binomial distribution, which models the number of successes in a fixed number of independent Bernoulli trials.

4
Q

What do probability distributions reveal?

A

Probability distributions show the likelihood of each outcome occurring in a random experiment or process.

5
Q

Can you provide an example of a uniform distribution?

A

Rolling a fair six-sided die has a uniform probability distribution.

6
Q

In what scenarios are probability distributions used?

A

Probability distributions are used to describe events or processes involving randomness.

7
Q

How are human heights modeled using distributions?

A

Human heights can be modeled using a normal distribution, with most people clustered around the average height.

8
Q

How do probability distributions assign probabilities?

A

Probability distributions assign specific probabilities to each potential outcome.

9
Q

Can you give an example of probability assignment?

A

Drawing a heart from a standard deck of cards has a probability of 1/4 (or 25%) because there are four suits, each equally likely.

10
Q

What is the purpose of analyzing uncertainty?

A

Probability distributions enable us to estimate the likelihood of different events and outcomes in a structured manner.

11
Q

What is an example of exclusive events?

A

Exclusive events are events that cannot happen simultaneously, meaning they are mutually exclusive. If one event occurs, the other cannot occur simultaneously. Example: Event A: Getting heads on a coin toss. Event B: Getting tails on a coin toss.

12
Q

Could you provide example of exclusive events?

A

Event C: Rolling an odd number on a six-sided die. Event D: Rolling an even number on a six-sided die.

13
Q

How can the probability of exclusive events be calculated?

A

For two exclusive events A and B, the probability of either event A or event B occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B)

14
Q

C What are inclusive events and what’s an example?

A

Inclusive events are events that can happen simultaneously, either partially or completely. They may have overlapping outcomes. Example: Event E: It rains. Event F: It is cloudy.

15
Q

Can you provide example of inclusive events?

A

Event G: Having an ice cream cone. Event H: Wearing sunglasses.

16
Q

How is the probability of inclusive events calculated?

A

For two inclusive events E and F, the probability of event E or event F occurring is the sum of their individual probabilities minus the probability of both events occurring: P(E or F) = P(E) + P(F) - P(E and F) Note: The term “P(E and F)” represents the probability of both events E and F happening together.

17
Q

What are independent events?

A

Independent events are events where the occurrence of one event does not affect the probability of the occurrence of another event. In other words, the outcome of one event has no influence on the outcome of the other event. Example: Event A: Flipping a coin and getting heads. Event B: Rolling a die and getting a 4.

18
Q

Could you provide another example of independent events?

A

Event C: Choosing a card from a deck and getting a heart. Event D: Tossing a fair coin and getting heads.

19
Q

How can the probability of independent events be calculated?

A

For two independent events A and B, the probability of both events occurring is the product of their individual probabilities: P(A and B) = P(A) × P(B)

20
Q

What are dependent events and what’s an example?

A

Dependent events are events where the occurrence of one event affects the probability of the occurrence of another event. The outcome of the first event influences the outcome of the second event. Example: Event E: Drawing a colored marble from a bag (without replacement). Event F: Drawing another marble of a different color from the same bag.

21
Q

Could you provide another example of dependent events?

A

Event G: Selecting a card from a deck and not replacing it. Event H: Selecting a second card from the deck after the first card was drawn.

22
Q

How is the probability of dependent events calculated?

A

For two dependent events A and B, the probability of both events occurring is the product of the probability of the first event and the conditional probability of the second event given that the first event has occurred: P(A and B) = P(A) × P(B|A)

23
Q

What is joint probability?

A

Joint probability refers to the probability of the intersection of two or more events, calculating the likelihood of both events occurring together.

24
Q

Could you provide an example of joint probability?

A

Let’s consider two events related to rolling a fair six-sided die: Event A: Rolling an even number (2, 4, or 6). Event B: Rolling a number greater than 3 (4, 5, or 6).

25
Q

How is joint probability calculated?

A

To calculate the joint probability of two events A and B, denoted as P(A and B), you can use the formula: P(A and B) = P(A) × P(B|A) Where: P(A) is the probability of event A occurring. P(B|A) is the conditional probability of event B occurring given that event A has occurred.

26
Q

What is joint probability?

A

Joint probability refers to the probability of two or more events occurring simultaneously or together. It calculates the likelihood of the intersection of events happening at the same time.

27
Q

C Could you provide an example of joint probability?

A

Certainly! Let’s consider two events related to drawing cards from a deck: Event A: Drawing a red card. Event B: Drawing a face card (jack, queen, or king).

28
Q

Calculate the join probability for
Example:
Let’s consider a dataset that represents the outcome of a survey where people were asked about their preferred mode of transportation to work. We’ll define two variables:

Variable X: Preferred mode of transportation (Car, Bike, Walk)
Variable Y: Weather condition (Sunny, Rainy)
Here’s a simplified version of the dataset:

A

Calculating Joint Probabilities:
To calculate the joint probability of specific combinations, divide the number of occurrences of the desired combination by the total number of observations.

For instance, let’s find the joint probability of people who prefer cars (X = Car) and the weather being sunny (Y = Sunny):

Number of times X = Car and Y = Sunny: 3 (from the dataset)
Total number of observations: 6

Joint Probability P(X = Car and Y = Sunny) = 3 / 6 = 0.5

Output Table:

Here’s a table showing the joint probabilities of the preferred mode of transportation and the weather condition:
Interpretation:
In the context of this dataset, the joint probability table indicates the likelihood of specific combinations of transportation preferences and weather conditions. For instance, the probability of a person preferring a car and the weather being sunny is 0.5.

Significance:
Joint probabilities allow us to analyze the co-occurrence of specific outcomes for two variables. They help identify patterns, dependencies, or associations between variables in real-world scenarios. In this example, we gain insights into how people’s preferred modes of transportation correspond to different weather conditions based on the joint probabilities calculated from the dataset.

29
Q

What are dependent events?

A

Dependent events are events where the occurrence of one event affects the probability of the occurrence of another event. The outcome of the first event influences the outcome of the second event.

30
Q

Can you give an example of dependent events?

A

Certainly! Let’s consider drawing cards from a deck again: Event C: Drawing a heart. Event D: Drawing a black card.

31
Q

How is the concept of dependent events related to joint probability?

A

In the context of dependent events, the probability of both events occurring together (joint probability) can be calculated by multiplying the probability of the first event by the conditional probability of the second event given that the first event has occurred.

32
Q

Can you provide an example that combines joint probability and dependent events?

A

Certainly! Consider drawing cards from a deck: Event E: Drawing an ace. Event F: Drawing a spade. To calculate the joint probability of drawing an ace and a spade, you would use the formula: P(E and F) = P(E) × P(F|E) Where: P(E) is the probability of drawing an ace. P(F|E) is the conditional probability of drawing a spade given that an ace has been drawn.

33
Q

C Are the formulas for joint probability and dependent events the same?

A

Yes, the formulas are the same. For both joint probability and dependent events, the formula is: P(A and B) = P(A) × P(B|A) However, dependent events specifically refer to situations where one event affects the probability of another event, while joint probability is a broader concept that considers the simultaneous occurrence of events, whether dependent or independent.

34
Q

What is a random variable?

A

A random variable is a mathematical concept used in probability and statistics to represent uncertain or random quantities. It assigns a numerical value to each possible outcome of a random process or experiment.

35
Q

Could you provide an example of a random variable?

A

Certainly! Let’s consider rolling a fair six-sided die. We can define a random variable “X” to represent the outcome of the roll: X = 1, if the die shows 1 X = 2, if the die shows 2 … X = 6, if the die shows 6

36
Q

How would you represent the random variable “X” for the die roll?

A

We can define the random variable “X” as follows: X = 1, if the die shows 1 X = 2, if the die shows 2 … X = 6, if the die shows 6

37
Q

What purpose does the random variable “X” serve in this example?

A

In this example, the random variable “X” helps us quantify and analyze the results of the random process of rolling the die. It provides a numerical representation for each possible outcome, allowing us to calculate probabilities, expected values, and other statistical measures based on this random variable.

38
Q

Are there different types of random variables?

A

Yes, random variables can be classified into two main types: discrete and continuous. In the example above, “X” is a discrete random variable because it represents a finite set of distinct values (the numbers on the die). Continuous random variables represent a range of possible values within a continuous interval.

39
Q

How are random variables used in probabilistic analysis and statistics?

A

Random variables are fundamental tools in probabilistic analysis and statistics. They enable us to model uncertain events, calculate probabilities of different outcomes, and make informed decisions based on data.

40
Q

What is marginal probability?

A

Marginal probability refers to the probability of a single event occurring, without considering the influence of any other variables.

41
Q

How do you calculate the marginal probability of each variable using this example and how is the formula applied in the example of marginal probability of “Gender”?

A

A1: To calculate the marginal probability of each variable, we sum up the probabilities of each category separately.

Marginal Probability of Gender:

P(Gender = Male) = 4/7 ≈ 0.571
P(Gender = Female) = 3/7 ≈ 0.429
Marginal Probability of Smoking Status:

P(Smoking Status = Smoker) = 3/7 ≈ 0.429
P(Smoking Status = Non-Smoker) = 4/7 ≈ 0.571

A2: In the example, when calculating the marginal probability of “Gender,” we sum the joint probabilities of “Gender” and “Smoking Status” for each gender category:

P(Gender = Male) = P(Gender = Male, Smoking Status) + P(Gender = Male, Non-Smoking Status) = (2/7) + (2/7) = 4/7

42
Q

Could you provide the formula for calculating marginal probability?

A

The formula for marginal probability is: P(X) = Σ P(X, Y=y)

43
Q

What does the formula P(X) = Σ P(X, Y=y) represent?

A

The formula P(X) = Σ P(X, Y=y) represents the calculation of the marginal probability of variable X by summing the joint probabilities of variables X and Y for each Y value.

44
Q

What is conditional probability?

A

Conditional probability calculates the likelihood of an event occurring given that another event has already occurred.

45
Q

Can you provide a formula for calculating conditional probability?

A
46
Q

Could you explain conditional probability with an example?

A

Let’s consider the example of rolling a fair six-sided die. Probability of rolling 2 given you rolled even num

47
Q

Bernoulli Distribution

A

Models binary events with two outcomes. Example: Flipping a biased coin.

48
Q

Binomial Distribution

A

Models successes in fixed independent trials. Example: Getting 3 heads in 10 coin flips.

49
Q

Poisson Distribution

A

Models counts in fixed intervals with an average rate. Example: Modeling customer arrivals with rate 5 per hour.

50
Q

Uniform Distribution

A

Constant probability density over an interval. Example: Modeling random numbers between 0 and 1.

51
Q

Normal Distribution

A

Bell-shaped distribution, widely used. Example: Modeling heights with mean 175 cm and st. dev. 6 cm.

52
Q

Exponential Distribution

A

Models time between events in a process. Example: Modeling time between phone calls with avg. 10 mins.

53
Q

Multinomial Distribution

A

Extends binomial to multiple categories. Example: Modeling outcomes of rolling a die multiple times.

54
Q

Multivariate Normal Distribution

A

Models correlated variables’ joint distribution. Example: Modeling portfolio of two stocks’ returns.

55
Q

What is a discrete probability distribution?

A

A discrete distribution assigns probabilities to distinct values and uses a Probability Mass Function (PMF).

56
Q

Example of a discrete distribution?

A

The Binomial distribution models successes in trials, like getting 3 heads in 10 coin flips.

57
Q

What is a continuous probability distribution?

A

A continuous distribution uses a Probability Density Function (PDF) to describe the likelihood within intervals.

58
Q

Example of a continuous distribution?

A

The Normal (Gaussian) distribution describes phenomena like human heights using a bell-shaped PDF.

59
Q

What is the Probability Mass Function (PMF)?

A

The PMF, used in discrete distributions, assigns probabilities to specific values of the random variable.

60
Q

Example illustrating PMF?

A

In a Bernoulli distribution, the PMF assigns probabilities to getting heads or tails when flipping a coin.

61
Q

What is the Probability Density Function (PDF)?

A

The PDF, in continuous distributions, describes the relative likelihood of a random variable falling within an interval.

62
Q

Example involving PDF?

A

The Normal distribution’s PDF describes the likelihood of observing a certain height within a given range.

63
Q

What are the mean and variance of a distribution?

A

The mean is the average value of a random variable, and variance measures the spread of values around the mean.

64
Q

How are probability distributions used?

A

Probability distributions model uncertainty, aiding predictions. For instance, the Poisson distribution predicts counts.

65
Q

Another application of probability distributions?

A

The Exponential distribution models time between events, optimizing resource allocation, like in call centers.

66
Q

Explain marginal probability with example dataset

A

Calculating Marginal Probabilities:

Let’s calculate the marginal probabilities of “Education Level” and “Employment Status” using this dataset.

Marginal Probability of Education Level:

P(Education Level = High School) = 3/7 ≈ 0.429
P(Education Level = College) = 2/7 ≈ 0.286
P(Education Level = Graduate) = 2/7 ≈ 0.286
Marginal Probability of Employment Status:

P(Employment Status = Employed) = 4/7 ≈ 0.571
P(Employment Status = Unemployed) = 3/7 ≈ 0.429
Output Table:

Here’s a table illustrating the marginal probabilities:

67
Q

Explain conditional probabilities with dataset

A

In the output table, we have calculated the conditional probabilities of going for a hike given different weather conditions.

Interpretation:

On sunny days, the probability of going for a hike is relatively high (0.75), indicating that people are more likely to go hiking when the weather is sunny.
On cloudy days, the probability of going for a hike is moderate (0.2857).
On rainy days, the probability of going for a hike is lower (0.2), suggesting that people are less likely to go hiking in rainy weather.
This example illustrates how conditional probabilities can help us understand the relationship between two events (in this case, weather conditions and outdoor activities) and make informed decisions or predictions based on the given data.