INTRODUCTION – Probability
The basic principles of probability will be learned through this module, which serves to prepare people practically for their application in data analysis. The emphasis will be on the basic probability rules pertaining to a single event, from which learners will build a basic theoretical formation for themselves in the first part of the module. It is expected that participants will move on later in the module to the advanced methods currently used by data professionals-the study of Bayes’ theorem, to mention one. This great tool used to deal with some complicated event types provides a relatively few resources in getting below the surface of probability within a context of data.
Core aspect of the newly modules involved a thorough study of probability distributions such as binomial, Poisson, and normal distributions. Participants will derive real structures of these distributions and learn to use them to find their patterns in data sets. At the end of this module, the participants will be quite well versed in principles and advanced techniques of probability so that they would apply these to confidently interpret and analyze the data.
Learning Objectives
- Model data in Python using probability distributions
- Explain z-scores and their importance in data analysis
- Define and apply the Empirical Rule
- Describe properties and applications of continuous probability distributions, including the normal distribution
- Explore specific discrete probability distributions, including the binomial and Poisson distributions
- Describe discrete or continuous random variables
- Explain Bayes’ theorem and its practical applications.
- Define dependent events and analyze these relationships.
- Describe conditional probability and its relevance in real-world scenarios.
- Classify event types, including mutually exclusive and independent events.
- Apply fundamental probability rules.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: BASIC CONCEPTS OF PROBABILITY
1. Objective probability is based on personal feeling, experience, or judgment.
- True
- False (CORRECT)
Correct: Such probability is subjective since one derives it from belief, experience, or intuition instead of empirical evidence. One generally applies this type of probability in situations where no statistical data is available and an individual must make decisions based on personal judgment or expertise.
2. Fill in the blank: In statistics, a number between _____ is used to express the probability that an event will occur.
- -1 and 1
- 0 and 1 (CORRECT)
- -1 and 0 1 and 2
Correct: A event’s probability is a number ranging from 0 to 1. Probabilities of 0 mean that the event cannot happen, while probabilities of 1 mean that it will certainly occur.
3. The probability of no snow tomorrow equals 1 minus the probability of snow tomorrow. This is an example of what rule of probability?
- Division rule
- Complement rule (CORRECT)
- Multiplication rule
- Addition rule
Correct: This definition gives the complement rule that the probability of event A not occurring equals 1 minus the probability of event A. The complement of an event in statistics indicates the condition of the event not happening.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: CONDITIONAL PROBABILITY
1. What is conditional probability?
- The probability of two events occurring at the same time
- The probability of an event occurring given that another event has already occurred (CORRECT)
- The probability of a single random event occurring
- The probability of a highly unlikely event occurring
Correct: The probability of an event, is a conditional probability event e given another event d has happened.
2. Suppose two events occur: The first event is drawing an ace from a standard deck of playing cards, and the second event is drawing another ace from the same deck. Note that the first ace is not reinserted into the deck after it is drawn. What term is used to describe these two events?
- Subjective
- Dependent (CORRECT)
- Objective
- Independent
Correct: Dependent events here ‘an event, event A followed by another event, event B’ is such that event A changes the outcome or probability of event B. The case in which both events take place is an illustration of dependent events where both events occur.
3. Fill in the blank: _____ probability is the updated probability of an event based on new data.
- Empirical
- Classical
- Posterior (CORRECT)
- Prior
Correct: The posterior probability is the updated probability of an event when incorporating new evidence or data, and the method of getting the posterior is using Bayes’ theorem to update the prior probability.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: DISCRETE PROBABILITY DISTRIBUTIONS
1. Which of the following statements describe continuous random variables? Select all that apply.
- Continuous random variables are typically whole numbers.
- Continuous random variables are typically negative numbers;
- Continuous random variables are typically decimal values. (CORRECT)
- Continuous random variables take all the possible values in some range of numbers. (CORRECT)
Correct: Continuous random variables may take any possible value within a numerical range. These are typically measurable quantities like height, weight, or time and are mostly quoted in decimals.
2. What probability distribution represents experiments with repeated trials that each have two possible outcomes: success or failure?
- The trinomial distribution
- The Poisson distribution
- The binomial distribution (CORRECT)
- The normal distribution
Correct: Binomial distribution is a probability distribution that inquires into experiments made up of several trials, each displaying an output of one of two options that is a success or failure.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: CONTINUOUS PROBABILITY DISTRIBUTIONS
1. The normal distribution has which of the following features? Select all that apply.
- The total area under the curve equals 4
- The shape is a bell curve (CORRECT)
- The curve is symmetrical on both sides of the center (CORRECT)
- The mean is located at the center of the curve (CORRECT)
Correct: This is a normal distribution: whose shape appears as a bell with a mean at the center of this bell and with symmetry on either side of the mean. This is the most popular probability distribution in delivering statistics because several types of data come with a bell-shaped pattern.
2. What does the empirical rule state?
- For a dataset with a normal distribution, 68% of values fall within 1 standard deviation of the mean, 95% of values fall within 2 standard deviations of the mean, and 99.7% of values fall within 3 standard deviations of the mean. (CORRECT)
- For a dataset with a normal distribution, 50% of values fall within 1 standard deviation of the mean, 30% of values fall within 2 standard deviations of the mean, and 20% of values fall within 3 standard deviations of the mean.
- For a dataset with a normal distribution, 100% of values fall within 1 standard deviation of the mean.
- For a dataset with a normal distribution, 33.3% of values fall within 1 standard deviation of the mean, 33.3% of values fall within 2 standard deviations of the mean, and 33.3% of values fall within 3 standard deviations of the mean.
Correct: As per the empirical rule, for any normally distributed data set, 68% of the values fall within one standard deviation of the mean, 95% of the values fall within two standard deviations of the mean, and 99.7% of the values fall within three standard deviations.
3. A data value is 2 standard deviations above the mean. What is its z-score?
- 0
- -2
- 2 (CORRECT)
- 1
Correct: It signifies that the data point is 2 standard deviations above the mean when it is given a z-score of 2. The z-score is the measure of how many standard deviations above or below the population mean a data point is.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: PROBABILITY DISTRIBUTIONS WITH PYTHON
1. A data professional is working with a dataset that has a normal distribution. To test out the empirical rule, they want to find out if roughly 68% of the data values fall within 1 standard deviation of the mean. What Python functions will enable them to compute the mean and standard deviation?
- mn() and std()
- mn() and stand()
- mean() and standard()
- mean() and std() (CORRECT)
Correct: For calculating the mean, we can make use of the mean() function, while to calculate standard deviation, we can use the std() function.
2. What Python function is used to compute z-scores for data?
- stats.zscore() (CORRECT)
- mean.zscore()
- median.zscore()
- normal.zscore()
Correct: The function stats.zscore() computes the z-scores of the data by using the Python language. This function comes under the scope of the stats module available in the package SciPy.
QUIZ: MODULE 2 CHALLENGE
1. A data professional is working for a large corporation. The marketing team asks them to predict the success of a new ad campaign. To make an informed prediction, they use statistics to analyze data on past ad campaigns. What type of probability are they using?
- Dependent
- Independent
- Objective (CORRECT)
- Subjective
2. The probability of an event is close to 1. Which of the following statements best describes the likelihood that the event will occur
- The event is unlikely to occur.
- The event is certain to occur.
- The event is certain not to occur.
- The event is likely to occur. (CORRECT)
3. The probability of rain tomorrow is 40%. What is the probability of the complement of this event?
- The probability of no rain tomorrow is 80%.
- The probability of no rain tomorrow is 20%.
- The probability of no rain tomorrow is 60%. (CORRECT)
- The probability of no rain tomorrow is 40%.
4. Fill in the blank: Two events are _____ if the occurrence of one event does not change the probability of the other event.
- continuous
- independent (CORRECT)
- discrete
- dependent
5. Fill in the blank: To calculate posterior probability, a data professional can use _____ to update the prior probability based on the data.
- the normal distribution
- Bayes’s theorem (CORRECT)
- the binomial distribution
- the complement rule
6. Which of the following statements accurately describes a key difference between discrete and continuous random variables?
- Discrete random variables are typically decimal values that can be measured; continuous random variables are typically whole numbers that can be counted.
- Discrete random variables are typically whole numbers that can be counted; continuous random variables are typically decimal values that can be measured. (CORRECT)
- Discrete random variables are positive numbers; continuous random variables are negative numbers.
- Discrete random variables are negative numbers; continuous random variables are positive numbers.*
7. The Poisson distribution can model which of the following kinds of data? Select all that apply.
- The number of heads in 10 fair coin tosses
- The number of calls per hour at a call center (CORRECT)
- The number of visitors per day on a website (CORRECT)
- The number of customers per week at a retail store (CORRECT)
8. A data professional working for a smartphone manufacturer is analyzing sample data on the weight of a specific smartphone. The data follows a normal distribution, with a mean weight of 150g and a standard deviation of 10g. According to the empirical rule, approximately what percentage of the data values lie between 140g and 160g?
- 95%
- 50%
- 68% (CORRECT)
- 99.7%
9. A data value has a z-score of 2.5. Where is it located?
- 2.5 standard deviations below the median
- 2.5 standard deviations above the median
- 2.5 standard deviations below the mean
- 2.5 standard deviations above the mean (CORRECT)
10. A data analytics team at a water utility works with a dataset that contains information about local reservoirs. They determine that the data follows a normal distribution. What Python function can they use to compute z-scores for the data?
- mean.zscore()
- describe()
- median.zscore()
- stats.zscore() (CORRECT)
11. Fill in the blank: The _____ distribution best models the number of heads in 10 fair coin flips.
- Bernoulli
- Poisson
- Binomial (CORRECT)
- Normal
12. If all outcomes of an event are equally likely, how is its probability calculated?
- Divide the number of desired outcomes by the total number of possible outcomes. (CORRECT)
- Divide the total number of possible outcomes by the number of desired outcomes.
- Divide the total number of certain outcomes by the number of possible outcomes.
- Divide the total number of possible outcomes by the number of certain outcomes.
13. A coin is tossed twice. To calculate the probability of getting two heads in a row, which of the following equations should be used?
- ½ ÷ ½
- ½ * ½ (CORRECT)
- ½ + ½
- ½ – ½
14. Which of the following events are mutually exclusive? Select all that apply.
- Getting heads on a first coin toss and tails on a second coin toss
- Getting a 4 on a first die roll and a 6 on a second die roll
- Getting heads and tails on the same coin toss (CORRECT)
- Getting a 4 and a 6 on the same die roll (CORRECT)
15. What concept refers to the probability of an event before new data is collected?
- Prior probability (CORRECT)
- Subjective probability
- Conditional probability
- Posterior probability
16. Which of the following are examples of continuous random variables? Select all that apply.
- The number of students in a math class
- The height of a redwood tree (CORRECT)
- The time it takes for a person to run a race (CORRECT)
- The weight of a polar bear (CORRECT)
17. A data professional working for a smartphone manufacturer is analyzing sample data on the weight of a specific smartphone. The data follows a normal distribution, with a mean weight of 150g and a standard deviation of 10g. What data value lies 3 standard deviations below the mean?
- 160g
- 120g (CORRECT)
- 130g
- 180g
18. The mean and the standard deviation of a standard normal distribution always equal what values?
- Mean = 2; standard deviation = 1
- Mean = 0; standard deviation = 2
- Mean = 1; standard deviation = 2
- Mean = 0; standard deviation = 1 (CORRECT)
19. A data professional is analyzing sales data for a retail store. The data follows a normal distribution. What Python function can they use to compute z-scores for the data?
- stats.zscore() (CORRECT)
- median.zscore()
- mean.zscore()
- normal.zscore()
20. A first coin toss results in tails, and a second coin toss results in heads. What concept best describes these two events?
- Subjective
- Non-random
- Independent (CORRECT)
- Dependent
21. What concept refers to the probability of an event occurring given that another event has already occurred?
- Classical probability
- Conditional probability (CORRECT)
- Subjective probability
- Empirical probability
22. Which of the following are examples of discrete random variables? Select all that apply.
- The length of an airplane
- The time it takes to drive from one city to another city
- The number of radios produced in a factory each day (CORRECT)
- The number of rooms in a hotel (CORRECT)
23. What probability distribution can model the probability of getting a certain number of defective products in a sample of 15 products?
- Binomial distribution (CORRECT)
- Normal distribution
- Standard normal distribution
- Poisson distribution
24. If a data value has a z-score of 0, what does the value equal?
- The median
- The standard deviation
- The mean (CORRECT)
- The mode
25. An investor believes there is a 90% chance that the price of a certain stock will increase in the next year. The investor’s prediction is based exclusively on intuition. What type of probability are they using?
- Subjective (CORRECT)
- Empirical
- Objective
- Classical
26. A six-sided die is rolled. To find the probability of rolling either a one or a three, what rule of probability should be used?
- Addition rule (CORRECT)
- Division rule
- Complement rule
- Multiplication rule
27. A jar contains four marbles: Two marbles are red, one is green, and one is blue. One marble is taken from the jar. What is the probability that the marble is blue?
- 100%
- 50%
- 25% (CORRECT)
- 75%
28. A data professional working for a smartphone manufacturer is analyzing sample data on the weight of a smartphone. The data follows a normal distribution, with a mean weight of 150g and a standard deviation of 10g. What data value lies at the center of the distribution curve?
- 160g
- 140g
- 10g
- 150g (CORRECT)
29. If the probability of an event equals 1, what is the chance that the event will occur?
- 1%
- 10%
- 50%
- 100% (CORRECT)
Correct: The function stats.zscore() in Python calculates z-scores of data. This is within the stats module that comes with the SciPy package.
30. Fill in the blank: The addition rule states that, if the events A and B are ____, then the probability of A or B happening is the sum of the probabilities of A and B.
- mutually inclusive
- mutually exclusive (CORRECT)
- highly likely
- highly unlikely
Correct: Absolutely! The addition rule is applied for mutually exclusive events in probability. Such events cannot occur simultaneously, for example, while flipping coins, heads and tails are mutually exclusive because both cannot occur at the same time.
31. Fill in the blank: Two events are _____ if the occurrence of one event changes the probability of the other event.
- independent
- dependent (CORRECT)
- subjective
- objective
Correct: Two events are dependent if the occurrence of one event changes the probability of the other event.
32. What does Bayes’s theorem enable data professionals to calculate?
- Interquartile range
- Standard deviation
- Mean
- Posterior probability (CORRECT)
Correct: Bayes’s theorem enables data professionals to calculate posterior probability, or the updated probability of an event based on new data.
33. Fill in the blank: A _____ random variable has a countable number of possible values.
- classical
- subjective
- discrete (CORRECT)
- continuous
Correct: Discrete random variables describe a finite number of values or a set of infinitely many values that can be counted.
34. Fill in the blank: The binomial distribution models the probability of events with _____ possible outcomes.
- four
- two (CORRECT)
- five
- three
Correct: The binomial distribution theoretically portrays a figure that is related to the probabilities of events with only two outcomes.
35. The Poisson distribution can model the probability that a certain number of events will occur during a specific time period.
- True (CORRECT)
- False
Correct: The Poisson distribution can model the probability that a certain number of events will occur during a specific time period.
36. What shape is the graph of a normal distribution?
- Triangular
- Rectangular
- Bell-shaped (CORRECT)
- Square
Correct: being interpreted such as: A normal distribution contains a continuous probability distribution that is symmetrical about the mean and has a shape that resembles that of a bell. It is fundamentally called “the bell curve” because of its form-an upward peak and two-sloping downward tails.
37. What is the z-score of a data value equal to the mean?
- 2
- 1
- 0 (CORRECT)
- 3
Correct: When the data value exactly equals the average, the z-score equals 0. This indicates how many standard deviations above or below the mean of the population the data point lies.
CONCLUSION – probability
This particular module delves deeply into probability and sets a strong basis for how to handle data analysis later. They learn the basics of single-event probability rules, then proceed onto more complex concepts such as Bayes theorem, while learning all along to have a diversity of tools for probabilistic analysis. Studies in probability distributions-such as the binomial, Poisson, or normal-further strengthen one’s analytical abilities, thereby increasing one’s understanding of the structure underlying the data.
Upon completing the module, students not only understand the theory of probability; they also gain practical experience that allows them to use these concepts in a variety of situations with different datasets. Real-life intervention ensures that this module is preparing participants for translating such complex events into action for making data-driven decisions or contributing significantly to the tasks of data analysis. After this training-in-depth on probability and its applications-participants will now be ready in the real field for the most common data interpretation and analysis challenges.