Discrete Probability and the Binomial Distribution

An introduction to probability and probabilistic thinking

�Discrete versus Continuous Random Variables

Discrete Random Variables

• Have outcomes that can be counted
• Number of times heads comes up on a coin flip out of 10 total flips
• Number of cars arriving at a parking garage in one hour
• Number of participants falling within a category

Continuous Random Variables

• Have [theoretically] infinitely many outcomes
• Temperatures
• Commute times

�Terminology for Probability

�Approaches to Probability

• Classical / Frequentist approach
• The probability of an event is the relative frequency of the event when the experiment is conducted for a very large number of trials
• Empirical approach
• The probability of an event is the number of times the event has been observed, divided by the number of times the experiment has been run
• Subjective / Belief-based approach
• The probability of an event is our educated belief that the event may occur (for example, the belief that our favorite sports team will win a championship this year)

�Estimating Probabilities: Example Scenarios

Determine each of the following (if possible) or describe the information you would need / use to do so…

Scenario 1: What is the probability that a coin flip results in “heads”?

Scenario 2: What is the probability that a manufactured item produced along this assembly line is defective?

Scenario 3: What is the probability that a randomly selected student is part of the Honors Program?

Scenario 4: What is the probability that a 6 or lower is flipped next at this Blackjack table?

�Probability Distributions: Probability Mass

�Uniform Probability and Calculations

�Contingency Tables and Probability

A contingency table lists the number of outcomes falling within categories

For example, the table below summarizes student involvement data.

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Excel Work: Open a new Excel Notebook and save it to your MAT240 folder as ProbabilityIntro.xlsx, copy/paste the two-way table into the first sheet

�Working with a Contingency Table

1. P(20+ hours)
2. P(at most one club)
3. P(20+ hours and one club)
4. P(No clubs or No work)
5. P(multiple clubs given not working)
6. Are being involved in no clubs and working 20+ hours independent?

Using your Excel spreadsheet, find the probabilities associated with a randomly chosen individual from this sample satisfying the events on the right.

�Check-In…

Before We Move Forward, Ask Me Two Questions…

A Special Class of [non-Uniform] Discrete �Distributions: The Binomial Distribution

�EXAMPLES OF BINOMIAL DISTRIBUTIONS

�Identifying Binomial Experiments

Which of the following scenarios correspond to binomial experiments? Why?

1. It is estimated that 77% of people have been in at least one car accident in their lives. Researchers asked 48 randomly selected individuals whether they have ever been in a car accident.
2. A researcher surveys students one by one, asking if they agree with a policy. The researcher continues until exactly 50 students say they agree with the policy.
3. You are measuring the time until a machine fails. Each machine’s lifespan is recorded, but failures occur at unpredictable times based on complex factors such as wear and tear.
4. A factory has a defect rate of 3% in the products it manufactures. Inspectors randomly select and evaluate 100 products.

Warm-Up Scenarios: Which of the following involve a Binomial random variable?

Scenario: A social media influencer receives an average of 25 text messages per hour. The number of messages they receive in a given hour is of interest.

Scenario: A group of birdwatchers is looking for a rare species of bird in a national park. Each time they hear a bird call, there’s a 5% chance it belongs to the rare species. They hear 50 bird calls during the day. The variable of interest is the number of calls belonging to the rare bird.

Scenario: An insurance company is interested in the number of minor car crashes a person has before they finally file an insurance claim.

Scenario: In an online multiplayer battle game, a particular player has a 40% chance of winning each match. During a weekend gaming tournament, the player participates in 15 matches and records how many they win.

Scenario: A tech company is analyzing the performance of its video streaming platform. They are interested in the amount of time between errors on the platform.

Calculating Probabilities with the Binomial Distribution

�Completed Example: Car Accidents, Part I

�Completed Example: Car Accidents, Part II

�Completed Example: Car Accidents, Part III

�Completed Example: Car Accidents, Part IV

�Example: Defective Products

Scenario: A factory has a defect rate of 3% in the products it manufactures. Inspectors randomly select 100 products. Find the probability of

• exactly 5 products having defects?
• at most 5 products having defects?
• at least 5 products having defects?

�Example: Growing plants from seeds

Scenario: A botanist finds that a certain species of plant successfully grows 60% of the time under controlled greenhouse conditions. The botanist plants 25 seeds.

• What is the probability that fewer than 20 of the seeds grow?
• What is the probability that at least 17 and no more than 21 of the seeds grow?
• What is the probability that more than 20 of the seeds grow?
• Should the botanist be surprised if fewer than 13 of the seeds grow?

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�Expected Value and Standard Deviation

�Example: Growing plants from seeds (Revisited)

Scenario: A botanist finds that a certain species of plant successfully grows 60% of the time under controlled greenhouse conditions. The botanist plants 25 seeds.

• Find the expected number of seeds to grow
• Find the standard deviation in the number of seeds to grow
• Does this change, weaken, strengthen, or confirm your response to whether or not the botanist should be surprised if fewer than 13 of the seeds grow?

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Exit Ticket

Navigate to our MAT240 Exit Ticket Form, answer the questions, and complete the task below.

Note. Today’s discussion is listed as 5. Discrete Probability

Task: Provide an example of a probability question that can be answered with basic (counting-based) techniques; solve the question you pose. Provide another example of a probability question that requires the use of the binomial distribution and describe why the binomial distribution is required for this scenario but not for the first.

If you can’t think of a context, a bag of colored marbles is a reasonable start.

�Summary

�Next Time…

• What we’ll be doing…
• Continuous Random Variables and the Normal Distribution

Homework: Complete the Topic 6 – Probability and the Normal Distribution interactive prep-work activity and submit both hash codes using the Google Form from this week’s BrightSpace announcement.