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Task List 03 — Probability Distributions (Selected Models)

Introduction

In many real-world random situations we are interested in the number of occurrences of certain events: defective products in a batch, the number of errors in a text, the waiting time until the first event occurs, or the number of requests in a system. To describe such phenomena mathematically, probability theory uses probabilistic models, also called probability distributions.

Each distribution describes a different type of random experiment and corresponds to different assumptions about how events occur.

In this task list we use five important models:

  • Binomial distribution (Bernoulli trials) – models the number of successes in a fixed number of independent trials with two outcomes (success/failure).
  • Hypergeometric distribution – used when we draw elements without replacement from a finite population containing elements of different types.
  • Geometric distribution – describes the number of trials needed to obtain the first success in a sequence of independent trials.
  • Poisson distribution – models the number of events occurring within a given time or space interval.
  • Multinomial distribution – a generalization of the binomial model when each trial can result in more than two possible outcomes.

The analysis of these models allows us to formally describe random experiments and compute probabilities of events in many practical situations.

Task 1 — Binomial Model (Quality Control)

In a factory, screws are produced. Each screw can be either good or defective.
The probability that a randomly selected screw is defective equals \(p\). Assume that successive inspections are independent and that the probability \(p\) is the same for each screw.

We consider an experiment consisting of checking 3 consecutive screws.

Tasks

  1. Describe the random experiment.
  2. Determine the sample space \( \Omega \).
  3. Specify the probabilities of the elements of the sample space.
  4. Define what is considered a success in this model.

Task 2 — Hypergeometric Model (Sampling from a Batch)

A warehouse contains 20 components, of which 5 are defective and 15 are functional.

We randomly select 4 components without replacement for inspection.

Tasks

  1. Describe the random experiment.
  2. Define the random variable \(X\) as the number of defective elements in the sample.
  3. Determine the possible values of \(X\).
  4. Provide the probability distribution.
  5. Explain what a success means in this model.

Task 3 — Geometric Model (Waiting for the First Event)

In a printing house, each printed page may contain a printing error with probability \(p\). Assume that pages are independent and that the probability \(p\) is the same for each page.

The experiment consists of observing consecutive pages until the first error appears.

Tasks

  1. Describe the random experiment.
  2. Determine the sample space \( \Omega \).
  3. Provide the probability distribution.
  4. Specify what is considered a success.

Task 4 — Poisson Model (Arrival of Events)

A web service receives on average 3 error reports per hour.

We assume that the number of reports in a given time interval follows a Poisson distribution.

Tasks

  1. Describe the random experiment.
  2. Determine the sample space \( \Omega \).
  3. Provide the formula of the probability distribution.
  4. Interpret the parameter \( \lambda \) and explain its value for one hour.

Task 5 — Multinomial Model (Categories of Outcomes)

A player rolls a die 5 times.

The outcomes are grouped into three categories:

  • small numbers (1–2)
  • medium numbers (3–4)
  • large numbers (5–6)

Thus, the probabilities of the three categories are \(1/3\), \(1/3\), and \(1/3\), respectively.

Tasks

  1. Describe the random experiment.
  2. Define the sample space.
  3. Specify the multinomial distribution.
  4. Explain the interpretation of the parameters.

Task 6 — Binomial Model

The probability of producing a defective part is 0.04. Assume that the inspections are independent.

An inspector checks 10 parts.

Calculate the probability that:

  • exactly 2 parts are defective,
  • at least one part is defective.

Task 7 — Hypergeometric Model

A box contains:

  • 12 working light bulbs
  • 3 defective ones

We draw 5 bulbs without replacement.

Calculate the probability that the sample contains exactly 2 defective bulbs.

Task 8 — Geometric Model

The probability of an error in program compilation is 0.1 for each compilation. Assume that successive compilations are independent.

A programmer performs consecutive compilations until the first error occurs.

Calculate the probability that:

  • the first error appears on the 4th compilation,
  • it appears no later than the 3rd compilation.

Task 9 — Poisson Model

A customer service center receives on average 5 requests per hour.

Calculate the probability that during one hour there will be:

  • exactly 3 requests,
  • at least one request.

Task 10 — Multinomial Model

A box contains candies of three flavors:

  • strawberry – 40%
  • lemon – 35%
  • mint – 25%

We perform 6 independent selections with replacement.

Calculate the probability that we obtain:

  • 3 strawberry
  • 2 lemon
  • 1 mint