Applications of Probability Distributions

Introduction

Probability distributions are mathematical models used to describe uncertainty and predict the likelihood of different outcomes.

They are used in many areas of everyday life, including business, medicine, engineering, environmental science, education, sports, and economics.

For example:

• A manufacturer may want to estimate the probability that a product is defective.

• A hospital may need to predict how many patients will arrive during an hour.

• A teacher may want to analyse examination results.

• A meteorologist may estimate the probability of extreme rainfall.

• A business may need to predict future sales.

• An insurance company may calculate the expected cost of claims.

Different situations require different probability models.

Choosing the correct distribution is essential because an inappropriate model can produce misleading conclusions, even when the calculations are mathematically correct.

Why Are Probability Distributions Useful?

A probability distribution describes the possible values of a random variable and the likelihood of those values occurring.

Probability distributions allow us to:

• Estimate the likelihood of particular events.

• Predict expected outcomes.

• Measure uncertainty and variability.

• Compare different decisions.

• Identify unusual observations.

• Evaluate risks.

• Make decisions using statistical evidence.

Example

A factory knows that approximately 4% of its products are defective.

Rather than inspecting every product, the factory may use a probability model to estimate the likelihood of finding defective items in a sample.

This information can help determine inspection procedures and quality-control standards.

Important Observation

Probability models do not guarantee particular outcomes.

They describe uncertainty and the likelihood of different possibilities.

Choosing an Appropriate Probability Model

The first step in solving a probability problem is identifying the type of random variable and the structure of the experiment.

Discrete Random Variables

Discrete random variables take countable values.

Examples include:

• Number of defective products.

• Number of successful attempts.

• Number of customers arriving.

• Number of telephone calls received.

Continuous Random Variables

Continuous random variables take values across numerical intervals.

Examples include:

• Height.

• Mass.

• Temperature.

• Time.

• Distance.

• Measurement error.

Common Probability Models

These distributions have different assumptions and mathematical properties.

A Guide to Selecting Probability Models

Important Questions

Before selecting a probability distribution, ask:

1. Is the variable discrete or continuous?

2. Are we counting events or measuring a quantity?

3. Is the number of trials fixed?

4. Are trials independent?

5. Does the probability of success remain constant?

6. Are events occurring at a constant average rate?

7. Is the observed distribution approximately symmetric?

8. Does the model reflect the real-world situation?

Bernoulli Distribution

A Bernoulli distribution describes a single experiment with two possible outcomes.

These outcomes are usually called success and failure.

Let:

X = 1 for success

X = 0 for failure

If the probability of success is p:

P(X = 1) = p

P(X = 0) = 1 − p

Example

A product has a 3% probability of being defective.

Let X = 1 if the product is defective and X = 0 otherwise.

Then:

P(X = 1) = 0.03

P(X = 0) = 0.97

Expected Value

E(X) = p

Variance

Var(X) = p(1 − p)

When Is This Model Appropriate?

Use a Bernoulli distribution when studying one trial with two possible outcomes.

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials with constant success probability.

Conditions

• Fixed number of trials.

• Two possible outcomes per trial.

• Independent trials.

• Constant probability of success.

Formula

P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ

where:

• n = number of trials

• k = number of successes

• p = probability of success

• C(n, k) = number of combinations

Expected Value

E(X) = np

Variance

Var(X) = np(1 − p)

Worked Example: Defective Products

A factory produces electronic components.

Each component has a 5% probability of being defective.

Ten independently produced components are inspected.

Find the probability that exactly two are defective.

Step 1: Identify the Distribution

X ~ Bin(10, 0.05)

n = 10

p = 0.05

k = 2

Step 2: Apply the Formula

P(X = 2) = C(10, 2)(0.05)²(0.95)⁸

C(10, 2) = 45

P(X = 2) = 45 × 0.0025 × 0.66342

P(X = 2) ≈ 0.0746

Interpretation

There is approximately a 7.46% probability that exactly two of the ten components will be defective.

Geometric Distribution

A geometric distribution models the number of independent trials required to obtain the first success when each trial has the same probability of success.

Example

A basketball player has a 70% probability of making each free throw.

Let X represent the number of attempts required to make the first successful shot.

Possible values:

X = 1, 2, 3, 4, ...

Formula

P(X = k) = (1 − p)ᵏ⁻¹p

where:

• p = probability of success

• k = trial number of the first success

Worked Example

Find the probability that the player makes the first successful shot on the third attempt.

p = 0.70

k = 3

P(X = 3) = (0.30)²(0.70)

P(X = 3) = 0.09 × 0.70

P(X = 3) = 0.063

Interpretation

There is a 6.3% probability that the player's first successful shot occurs on the third attempt.

Expected Value

For this definition of the geometric random variable:

E(X) = 1/p

For p = 0.70:

E(X) = 1/0.70

E(X) ≈ 1.43 attempts

Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time, distance, area, or another exposure measure.

It is useful when events occur independently at a constant average rate.

Examples include:

• Telephone calls received per minute.

• Customers arriving at a service desk.

• Radioactive decay events.

• Defects per metre of material.

• Equipment failures over time.

Poisson Formula

P(X = k) = e⁻λ λᵏ/k!

where:

• λ = expected number of events in the interval

• k = number of events

• e ≈ 2.71828

Expected Value

E(X) = λ

Variance

Var(X) = λ

Worked Example: Customer Arrivals

A shop receives an average of four customers every ten minutes.

Assume customer arrivals follow a Poisson process.

Find the probability that exactly three customers arrive during a ten-minute interval.

Step 1: Identify the Values

λ = 4

k = 3

Step 2: Apply the Formula

P(X = 3) = e⁻⁴ × 4³/3!

P(X = 3) = e⁻⁴ × 64/6

P(X = 3) ≈ 0.0183156 × 10.6667

P(X = 3) ≈ 0.1954

Interpretation

There is approximately a 19.54% probability that exactly three customers arrive during a ten-minute interval.

Uniform Distribution

A uniform distribution describes situations where outcomes have equal probabilities or equal probability density.

There are two main forms.

Discrete Uniform Distribution

A discrete uniform distribution has a finite number of equally likely outcomes.

Example

Rolling a fair six-sided die.

Each value has probability:

P(X = x) = 1/6

The expected value is:

E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6

E(X) = 3.5

Continuous Uniform Distribution

A continuous uniform distribution has constant probability density over an interval.

For X uniformly distributed between a and b:

f(x) = 1/(b − a)

for a ≤ x ≤ b.

Probability Formula

P(c ≤ X ≤ d) = (d − c)/(b − a)

provided a ≤ c ≤ d ≤ b.

Expected Value

E(X) = (a + b)/2

Worked Example: Waiting Time

A bus arrives at a random time during a 20-minute interval.

Assume every arrival time within the interval is equally likely.

Find the probability that a passenger waits between 5 and 12 minutes.

Step 1: Identify the Interval

Total interval = 20 minutes

Desired interval = 12 − 5 = 7 minutes

Step 2: Calculate the Probability

P(5 ≤ X ≤ 12) = 7/20

P(5 ≤ X ≤ 12) = 0.35

Interpretation

There is a 35% probability that the waiting time is between 5 and 12 minutes.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped.

It is commonly used to model:

• Measurement errors.

• Some biological measurements.

• Some examination results.

• Manufacturing measurements.

• Averages of large samples.

Characteristics

• Symmetric around the mean.

• Mean, median, and mode are equal.

• Total area under the curve is 1.

• Shape determined by mean and standard deviation.

Normal Distribution Notation

X ~ N(μ, σ²)

z-Score Formula

z = (x − μ)/σ

Worked Example: Examination Scores

Examination scores are approximately normally distributed with:

Mean = 70

Standard deviation = 10

Find the probability that a randomly selected student scores above 85.

Step 1: Calculate the z-Score

z = (85 − 70)/10

z = 1.5

Step 2: Find the Probability

P(Z ≤ 1.5) = 0.9332

P(Z > 1.5) = 1 − 0.9332

P(Z > 1.5) = 0.0668

Interpretation

Approximately 6.68% of students are expected to score above 85 under the normal model.

Exponential Distribution

The exponential distribution models the waiting time between events in a Poisson process.

Examples include:

• Time between customer arrivals.

• Time between certain equipment failures.

• Time between radioactive decay events.

• Time between calls received by a service centre.

Probability Formula

For a waiting time X with rate λ:

P(X > t) = e⁻λᵗ

where:

• λ = average event rate per unit time

• t = waiting time

Expected Value

E(X) = 1/λ

Worked Example

A service centre receives an average of three calls per hour.

Assume calls arrive according to a Poisson process.

Find the probability that the waiting time until the next call exceeds 20 minutes.

Step 1: Convert Time

20 minutes = 1/3 hour

Step 2: Identify the Rate

λ = 3 calls per hour

Step 3: Apply the Formula

P(X > 1/3) = e⁻³⁽¹/³⁾

P(X > 1/3) = e⁻¹

P(X > 1/3) ≈ 0.3679

Interpretation

There is approximately a 36.79% probability that the next call will arrive more than 20 minutes later.

Comparing Probability Distributions

Important Observation

Two models may describe different aspects of the same process.

For example, a Poisson distribution can describe the number of calls arriving during an hour.

An exponential distribution can describe the waiting time until the next call.

The appropriate model depends on the question being asked.

Solving Real-World Probability Problems

A reliable method for solving probability problems involves six steps.

Identify the Random Variable

Define exactly what is being measured or counted.

For example:

X = number of defective products.

Select the Distribution

Determine which probability model best describes the situation.

Check the Assumptions

Confirm that the conditions of the model are reasonable.

Identify the Parameters

Determine the necessary values, such as:

n, p, λ, μ, or σ.

Calculate the Probability

Apply the appropriate formula or statistical method.

Interpret the Result

Explain what the numerical answer means in the context of the original problem.

Example

A factory has a defect rate of 2%.

Twenty independently produced products are selected.

Find the probability that at least one product is defective.

Step 1: Define the Variable

X = number of defective products.

Step 2: Select the Model

Binomial distribution.

Step 3: Identify the Parameters

n = 20

p = 0.02

Step 4: Use the Complement Rule

P(X ≥ 1) = 1 − P(X = 0)

P(X ≥ 1) = 1 − (0.98)²⁰

P(X ≥ 1) ≈ 1 − 0.6676

P(X ≥ 1) ≈ 0.3324

Interpretation

There is approximately a 33.24% probability that at least one of the twenty products will be defective.

Expected Value in Decision-Making

Expected value is useful when comparing decisions involving uncertainty.

For a discrete random variable:

E(X) = Σ[x × P(X = x)]

Worked Example: Business Investment

A business considers investing in a new product.

The possible profits are:

Step 1: Calculate Weighted Outcomes

High demand:

20,000 × 0.30 = 6,000

Moderate demand:

8,000 × 0.50 = 4,000

Low demand:

−5,000 × 0.20 = −1,000

Step 2: Add

E(X) = 6,000 + 4,000 − 1,000

E(X) = $9,000

Interpretation

The expected profit is $9,000.

This does not guarantee a profit of $9,000.

The actual outcome may be a large profit, a moderate profit, or a loss.

Important Consideration

Expected value is not the only factor in decision-making.

Businesses may also consider:

• Financial risk.

• Available resources.

• Worst-case outcomes.

• Variability of returns.

• Long-term consequences.

Interpreting Statistical Results

A probability calculation is only useful when its meaning is correctly explained.

Probability Versus Certainty

A probability of 0.80 does not guarantee that an event will occur.

It means the event has an 80% probability under the model.

Expected Value Versus Actual Outcome

An expected value of 3.5 successes does not mean that 3.5 successes will occur in an individual experiment.

It represents a long-run average.

Statistical Models Versus Reality

Models simplify real-world situations.

A model may assume independence or constant probabilities even when real-world conditions are more complicated.

Example

A basketball player may have a historical free-throw success rate of 80%.

However, performance may change due to:

• Fatigue.

• Injury.

• Pressure.

• Training.

• Environmental conditions.

A binomial model may be useful, but its assumptions should be evaluated.

Evaluating Statistical Models

A statistical model should be assessed before its conclusions are accepted.

Accuracy

Does the model produce predictions that are reasonably consistent with observed data?

Assumptions

Are the assumptions realistic?

For example, are repeated trials truly independent?

Data Quality

Were the data collected using a reliable method?

Sample Size

Is there enough information to estimate model parameters reliably?

Outliers

Are unusual observations affecting the results?

Model Limitations

Does the model ignore important real-world factors?

Example: Customer Arrivals

A Poisson model assumes events occur independently at a constant average rate.

However, customer arrivals may be more frequent during lunchtime and less frequent during the evening.

A single constant-rate Poisson model may not describe the entire day accurately.

A better model might use different arrival rates for different time periods.

Comparing Model Predictions with Observed Data

One way to evaluate a model is to compare its predicted probabilities with observed frequencies.

Example

A factory predicts that 5% of products are defective.

During an inspection, 1,000 products are examined.

The expected number of defective products is:

E(X) = 1,000 × 0.05

E(X) = 50

Suppose the inspector finds 83 defective products.

Interpretation

The observed number is substantially higher than the expected value.

This may suggest that:

• The defect rate has increased.

• The original estimate was inaccurate.

• The sample was unusual.

• The independence or constant-probability assumptions are inappropriate.

Important Observation

A difference between observed and expected values does not automatically prove that the model is wrong.

Random variation must also be considered.

Formal statistical tests can help determine whether the difference is unusually large under the model.

Using Graphs to Evaluate Probability Models

Graphs can reveal whether a theoretical distribution matches observed data.

Histograms

Histograms show the shape of numerical data.

They can reveal:

• Symmetry.

• Skewness.

• Multiple peaks.

• Gaps.

• Outliers.

Probability Distribution Graphs

These show the theoretical probabilities assigned to possible outcomes.

Box Plots

Box plots summarise the median, quartiles, and spread of data.

Normal Probability Plots

Normal probability plots help assess whether data are reasonably consistent with a normal distribution.

Important Observation

A graph may suggest that a model is inappropriate, but visual inspection alone may not provide a complete evaluation.

Numerical summaries and statistical tests may also be useful.

Understanding Risk and Uncertainty

Probability models help quantify risk.

Risk involves the possibility of an undesirable outcome and its potential consequences.

Example

Consider two investments.

Investment A:

• Expected profit = $5,000.

• Small variability.

Investment B:

• Expected profit = $7,000.

• Large variability.

Investment B has a higher expected profit.

However, it may also have a greater probability of a substantial loss.

Important Observation

A higher expected value does not always make an option preferable.

Decision-makers must consider both expected outcomes and uncertainty.

Communicating Statistical Conclusions

Statistical conclusions should be clear, accurate, and connected to the original question.

A strong statistical conclusion should include:

• The probability model used.

• The relevant assumptions.

• The calculated result.

• An explanation of the result in context.

• Any important limitations.

Weak Conclusion

"The probability is 0.067."

This statement provides a number but little meaning.

Strong Conclusion

"Using a normal distribution with a mean of 70 and standard deviation of 10, the probability that a student scores above 85 is approximately 6.7%. This estimate assumes the examination scores are reasonably described by the normal model."

The stronger conclusion identifies the model, explains the result, and acknowledges an assumption.

Avoid Overstating Conclusions

Statistical results describe uncertainty.

Avoid statements such as:

• "This event will definitely happen."

• "The model proves the outcome."

• "Every sample will have the expected value."

Instead, use language such as:

• "The model predicts..."

• "The probability is approximately..."

• "The expected value is..."

• "The evidence suggests..."

• "Under the stated assumptions..."

Real-World Application: Healthcare

Healthcare researchers use probability distributions to model patient outcomes and service demand.

Example

A hospital estimates that 10% of patients require an additional follow-up appointment.

Suppose 50 independent patients are selected, each with the same probability.

Let X represent the number requiring follow-up.

X ~ Bin(50, 0.10)

Expected Value

E(X) = 50 × 0.10

E(X) = 5

Interpretation

The hospital expects approximately five patients in the group to require follow-up.

This estimate may help with staffing and appointment planning.

However, actual patient needs may vary.

Real-World Application: Environmental Science

Environmental scientists use probability models to study rainfall, pollution, temperature, and extreme events.

Example

Suppose the number of heavy rainfall events in a region follows a Poisson distribution with an average of two events per month.

Find the probability of no heavy rainfall events during a month.

Step 1: Identify the Values

λ = 2

k = 0

Step 2: Apply the Formula

P(X = 0) = e⁻² × 2⁰/0!

P(X = 0) = e⁻²

P(X = 0) ≈ 0.1353

Interpretation

There is approximately a 13.53% probability of no heavy rainfall events during a month under the Poisson model.

Actual weather patterns may violate the assumptions of a constant-rate process.

Real-World Application: Engineering

Engineers use probability models to study manufacturing tolerances, measurement errors, and equipment reliability.

Example

A machine produces metal rods with lengths approximately normally distributed.

Mean = 100 mm

Standard deviation = 0.5 mm

Find the probability that a rod has a length between 99 mm and 101 mm.

Step 1: Calculate z-Scores

Lower boundary:

z = (99 − 100)/0.5 = −2

Upper boundary:

z = (101 − 100)/0.5 = 2

Step 2: Calculate Probability

P(−2 ≤ Z ≤ 2) ≈ 0.9544

Interpretation

Approximately 95.44% of rods are expected to fall within the specified length interval.

This information can help engineers evaluate manufacturing consistency.

Common Mistakes

Choosing a distribution without checking its assumptions

A familiar formula is not necessarily the correct model.

Confusing discrete and continuous variables

Counts are generally discrete, while measurements are often continuous.

Using a binomial model when trials are dependent

Dependence can invalidate the standard binomial assumptions.

Using a Poisson model without considering the event rate

A changing event rate may require a different model.

Assuming every measurement is normally distributed

Real-world measurements may be skewed or multimodal.

Confusing expected value with the most likely outcome

The expected value and mode are not necessarily equal.

Confusing probability with certainty

A high probability does not guarantee an outcome.

Ignoring units

Probabilities are unitless, but expected values and standard deviations retain the units of the random variable.

Using the wrong parameter

Different distributions require different parameters.

Ignoring model limitations

Every statistical model simplifies reality.

Overinterpreting unusual observations

An unusual result does not automatically prove that a model is incorrect.

Communicating only a numerical answer

A useful conclusion explains the meaning of the result in context.

Key Terms

• Probability Distribution: Mathematical description of possible outcomes and their probabilities.

• Probability Model: Mathematical representation of uncertainty.

• Random Variable: Numerical value assigned to outcomes of a random experiment.

• Discrete Random Variable: Random variable with countable possible values.

• Continuous Random Variable: Random variable taking values across continuous intervals.

• Bernoulli Distribution: Distribution describing one success-or-failure trial.

• Binomial Distribution: Distribution describing successes in a fixed number of independent trials.

• Geometric Distribution: Distribution describing the number of trials until the first success.

• Poisson Distribution: Distribution describing event counts in fixed intervals under suitable assumptions.

• Uniform Distribution: Distribution with equally likely outcomes or constant density.

• Normal Distribution: Continuous symmetric bell-shaped probability distribution.

• Exponential Distribution: Distribution describing waiting times in a Poisson process.

• Expected Value: Probability-weighted average of a random variable.

• Variance: Measure of the spread of a probability distribution.

• Standard Deviation: Square root of variance.

• Parameter: Numerical value describing a probability model or population.

• Probability Density: Function describing probability per unit interval for continuous variables.

• Cumulative Probability: Probability of an outcome at or below a specified value.

• Independence: Condition where one event does not affect another's probability.

• Assumption: Condition required for a model or calculation to be appropriate.

• Model Accuracy: Degree to which a model agrees with observed data.

• Model Validation: Process of evaluating whether a model describes data adequately.

• Sampling Variability: Natural variation arising from random sampling.

• Statistical Inference: Use of sample evidence to draw conclusions about a population.

• Risk: Possibility and consequences of an undesirable outcome.

• Uncertainty: Lack of certainty about future or unknown outcomes.

Key Takeaways

• Probability distributions help describe uncertainty and predict outcomes.

• Different situations require different probability models.

• The type of random variable helps determine which distribution may be appropriate.

• Bernoulli distributions describe single binary trials.

• Binomial distributions describe numbers of successes in fixed independent trials.

• Geometric distributions describe trials until the first success.

• Poisson distributions describe event counts over intervals.

• Uniform distributions describe equally likely outcomes or constant density.

• Normal distributions describe approximately symmetric bell-shaped measurements.

• Exponential distributions describe waiting times in Poisson processes.

• Expected value describes the long-run average outcome.

• Variance and standard deviation describe variability.

• Model assumptions must be checked before calculations are interpreted.

• Observed data should be compared with model predictions.

• Statistical conclusions should include context, assumptions, and limitations.

• Probability models support decision-making but do not guarantee outcomes.

Check Your Understanding

Selecting Probability Models

1. Which distribution describes a single success-or-failure trial?

2. Which distribution models the number of successes in ten independent trials?

3. Which distribution models the number of trials until the first success?

4. Which distribution may model customer arrivals during an hour?

5. Which distribution is commonly used for symmetric bell-shaped measurements?

Solving Probability Problems

6. A fair coin is tossed five times. Which distribution is appropriate for the number of heads?

7. A fair die is rolled once. Which distribution describes the equally likely numerical outcomes?

8. A shop receives an average of three customers per minute. Which distribution may model the number arriving in one minute?

9. A bus arrives uniformly during a 15-minute interval. What is the probability of waiting less than five minutes?

10. A product has a 10% defect probability. What is the expected number of defective products in 50 independent items?

Interpreting Results

11. What does an expected value represent?

12. Why does a probability of 0.90 not guarantee an event?

13. Explain why model assumptions are important.

14. What is the difference between a model prediction and an observed result?

15. Why might a statistical model become less accurate over time?

Evaluating Models

16. Why might a binomial model be inappropriate for dependent trials?

17. Why might a Poisson model be inappropriate when arrival rates change throughout the day?

18. Explain why a normal model may be inappropriate for strongly skewed data.

19. Describe one method for comparing a model with observed data.

20. Identify two features of a clear statistical conclusion.

Answer Key

Extended Challenge

Selecting Appropriate Models

1. A factory inspects twenty independently produced products and counts defects. Identify the appropriate distribution.

2. A researcher measures the heights of 500 adults. Explain when a normal model may be appropriate.

3. A shop counts customer arrivals during ten-minute intervals. Explain when a Poisson model may be appropriate.

4. A player rolls a die until the first six appears. Identify the distribution.

5. A passenger waits for a bus that arrives uniformly during a 30-minute interval. Identify the distribution.

Probability Calculations

6. A fair coin is tossed six times. Find the probability of exactly three heads.

7. A product has a 4% defect rate. Find the probability of no defects in ten independent products.

8. A shop receives an average of five customers per hour according to a Poisson process. Find the probability of exactly three arrivals in an hour.

9. A normal distribution has mean 100 and standard deviation 15. Find the probability of an observation exceeding 130.

10. A waiting time is uniformly distributed between 0 and 20 minutes. Find the probability of waiting between 8 and 15 minutes.

Expected Value and Risk

11. A game awards $20 with probability 0.2 and nothing otherwise. Calculate the expected prize.

12. If the game costs $5, calculate the expected net gain.

13. Explain why a higher expected profit does not always indicate a better investment.

14. Describe how variance can help compare two risky decisions.

15. Explain why businesses use probability models when planning inventory.

Model Evaluation

16. A factory predicts 2% defective products but observes 8% in a large sample. Explain possible reasons for the difference.

17. Explain how a histogram can help evaluate a normal model.

18. Describe a situation where a model's independence assumption might fail.

19. Explain why a model should be updated when new data become available.

20. Write a short statistical conclusion that includes a calculated probability, its meaning, and a limitation.

Final Investigation: Choosing the Best Probability Model

A research team investigates four different situations.

Situation A: Manufacturing

A factory produces electronic components.

Each component has a 3% probability of being defective.

Twenty independently produced components are inspected.

Situation B: Customer Arrivals

A shop receives an average of six customers per hour.

Assume arrivals follow a Poisson process.

Situation C: Examination Results

Examination scores are approximately normally distributed.

Mean = 70

Standard deviation = 12

Situation D: Waiting Time

A bus arrives uniformly at random during a 20-minute interval.

Investigation Tasks

1. Identify the random variable in Situation A.

2. Select the appropriate probability distribution for Situation A.

3. Explain why the distribution is appropriate.

4. Calculate the probability of exactly one defective component.

5. Calculate the expected number of defective components.

6. Identify the random variable in Situation B.

7. Select the appropriate distribution for Situation B.

8. Calculate the probability of exactly four customers arriving in one hour.

9. Calculate the probability of no customers arriving in one hour.

10. Explain one limitation of the model in Situation B.

11. Identify the distribution used in Situation C.

12. Calculate the z-score for an examination score of 88.

13. Calculate the probability of scoring above 88.

14. Calculate the probability of scoring between 58 and 82.

15. Explain how the results could help a teacher interpret student performance.

16. Identify the distribution used in Situation D.

17. Calculate the probability of waiting less than five minutes.

18. Calculate the probability of waiting between 8 and 15 minutes.

19. Compare the assumptions and limitations of all four models.

20. Write a statistical report explaining how probability distributions support real-world decision-making.