Recording-2025-02-19T16:07:29.050Z

Understanding Blood Alcohol Content (BAC) in Drivers

Overview of the Problem

• A study examines the probability of 15 drivers having a blood alcohol content (BAC) exceeding the legal limit.

• 20% of all drivers are reported to have BAC levels above the limit, which translates to a probability (p) of 0.2.

Part A: Probability of All 15 Drivers Over the Limit

• Goal: Calculate the probability that all 15 drivers have a BAC level exceeding the legal limit.

• Probability of success (p) = 0.2

• Probability of failure (q) = 1 - p = 0.8

• The formula for the binomial probability is:

  • P(X = k) = (n choose k) * p^k * q^(n-k)

• For 15 drivers (k = 15):

  • P(X = 15) = (15 choose 15) * (0.2)^15 * (0.8)^0 = 1 * (0.2)^15

• This evaluates as

  • (0.2)^15 is a very small number, thus approximately equal to 0.

• Conclusion: The probability of all 15 drivers having BAC above the limit is essentially zero.

Part B: Probability of Exactly 6 Drivers Over the Limit

• Goal: Calculate the probability of exactly 6 drivers having BAC above the limit.

• Using the binomial probability formula for k = 6:

  • P(X = 6) = (15 choose 6) * (0.2)^6 * (0.8)^9

• Calculation results in:

  • Approximately 0.043 (or 4.3%).

• Conclusion: There is a 4.3% probability of exactly 6 drivers exceeding the BAC limit.

Part C: Probability of 6 or More Drivers Over the Limit

• Goal: Find P(X ≥ 6).

• This requires calculating the probabilities of X = 6, 7, ..., up to 15.

• Advice: Use statistical software or online binomial distribution calculators.

• Result obtained: Approximately 0.06.

• Conclusion: The probability of 6 or more drivers exceeding the legal BAC limit is about 6%.

Part D: Probability of 15 Drivers Within Legal Limit

• Goal: Determine the probability that all drivers are within the legal limit.

• This is equivalent to finding P(Y = 0), where Y is the number of drivers above the limit (success).

• Since p = 0.2 (probability of being over the limit), the probability of being within the legal limit is:

  • q = 0.8.

• Therefore, for Y = 15, we calculate:

  • P(X = 0) = (15 choose 0) * (0.2)^0 * (0.8)^15

• Conclusion: The probability of all 15 drivers being within the legal limit is calculated using the binomial formula and results in a significant value, indicating that it’s plausible for all drivers to be below the limit.

Part E: Mean and Standard Deviation

• Mean Calculation: Mean (μ) = n * p = 15 * 0.2 = 3.

• Standard Deviation Calculation: Standard deviation (σ) is found using:

  • σ = sqrt(n * p * (1-p)) = sqrt(15 * 0.2 * 0.8) ≈ 1.55.

• Interpretation: On average, 3 out of 15 drivers are expected to exceed the legal BAC limit, with a standard deviation of approximately 1.55, indicating variability in the number of drivers exceeding the limit.

Continuous Probability Distributions

Introduction to Continuous Random Variables

• Continuous random variables deal with measurements that can take on any value within a certain range, unlike discrete variables that count occurrences.

• Examples: Height, weight, GPA, distance.

Probability Density Function (PDF)

• Denoted as f(y).

• The area under the curve of f(y) equals 1.

• The PDF helps to determine probabilities over intervals rather than specific values.

Normal Distribution

• Characteristics: Bell-shaped and defined by mean (μ) and standard deviation (σ).

• Can be expressed as: Y ~ N(μ, σ) where N represents normal distribution.

• Standard normal distribution is centered at 0 with a standard deviation of 1.

Empirical Rule (68-95-99.7 Rule)

• Approximately 68% of values lie within one standard deviation of the mean.

• About 95% lie within two standard deviations.

• Approximately 99.7% lie within three standard deviations.

Z-Score Standardization

• Z-Score is calculated as: Z = (Y - μ) / σ.

• Z-scores help compare different datasets that may vary in scale but share normal distribution patterns.

• Provides insight into how far a specific value is from the mean in terms of standard deviations.