Probability – definition, rules and applications in pharmacy MCQs With Answer

Probability is a foundational concept in pharmacy that quantifies uncertainty in outcomes such as adverse drug reactions, assay variability, and clinical trial results. This introduction covers the definition of probability, core rules (axioms, complement, addition, multiplication, conditional probability, Bayes theorem) and practical applications in pharmaceutics, pharmacokinetics, quality control and diagnostics. Understanding probability distributions (binomial, Poisson, normal), expected value and variance helps B. Pharm students analyze experimental data, design sampling plans, interpret diagnostic tests (sensitivity, specificity, predictive values) and assess risk. These concepts directly support evidence-based decisions in formulation, QC and pharmacovigilance. ‘Now let’s test your knowledge with 30 MCQs on this topic.’

Q1. What is the classical definition of probability for an event A when all outcomes are equally likely?

• The ratio of favorable outcomes to total possible outcomes
• The long-run relative frequency of A after many trials
• A subjective degree of belief about A
• The expected value of A

Correct Answer: The ratio of favorable outcomes to total possible outcomes

Q2. Which of the following is NOT one of Kolmogorov’s three axioms of probability?

• Non-negativity: P(A) ≥ 0 for any event A
• Normalization: P(S) = 1 for the sample space S
• Finite additivity for disjoint events only
• Countable additivity for a sequence of disjoint events

Correct Answer: Finite additivity for disjoint events only

Q3. The complement rule for an event A is best stated as:

• P(A’) = 1 − P(A)
• P(A’) = P(A)/ (1 − P(A))
• P(A’) = P(A) + P(A and not A)
• P(A’) = 1 / P(A)

Correct Answer: P(A’) = 1 − P(A)

Q4. What is the correct addition rule for the probability of union of two events A and B?

• P(A ∪ B) = P(A) + P(B)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• P(A ∪ B) = P(A)P(B)
• P(A ∪ B) = P(A|B) + P(B|A)

Correct Answer: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Q5. For two independent events A and B, which formula gives the probability of both occurring?

• P(A ∩ B) = P(A) + P(B)
• P(A ∩ B) = P(A)P(B)
• P(A ∩ B) = P(A|B) − P(B|A)
• P(A ∩ B) = P(A) / P(B)

Correct Answer: P(A ∩ B) = P(A)P(B)

Q6. How is conditional probability P(A|B) defined?

• P(A|B) = P(A ∪ B) / P(B)
• P(A|B) = P(A) · P(B)
• P(A|B) = P(A ∩ B) / P(B)
• P(A|B) = P(B) / P(A ∩ B)

Correct Answer: P(A|B) = P(A ∩ B) / P(B)

Q7. Bayes theorem is used to compute which of the following?

• The probability of B given A using only P(B)
• The posterior probability P(A|B) from P(B|A), P(A) and P(B)
• The prior probability without new data
• The union probability of two mutually exclusive events

Correct Answer: The posterior probability P(A|B) from P(B|A), P(A) and P(B)

Q8. Two events A and B are independent if and only if:

• P(A ∩ B) = 0
• P(A ∩ B) = P(A) + P(B)
• P(A ∩ B) = P(A)P(B)
• P(A|B) = 0

Correct Answer: P(A ∩ B) = P(A)P(B)

Q9. The law of total probability helps compute P(B) when:

• B can be partitioned by mutually exclusive events A1, A2,… and P(B) = Σ P(B|Ai)P(Ai)
• B is independent of all Ai events
• B is a single simple event with known probability
• B occurs with probability one

Correct Answer: B can be partitioned by mutually exclusive events A1, A2,… and P(B) = Σ P(B|Ai)P(Ai)

Q10. For a discrete random variable X, the expected value E[X] is:

• The most probable outcome of X
• The variance of X
• The weighted average of possible values using their probabilities
• The maximum value X can take

Correct Answer: The weighted average of possible values using their probabilities

Q11. Variance Var(X) of a random variable X is defined as:

• E[X] − median(X)
• E[(X − E[X])^2]
• P(X > mean)
• The square root of the mean

Correct Answer: E[(X − E[X])^2]

Q12. In a binomial distribution with parameters n and p, the mean is:

• n + p
• n / p
• np
• n(1 − p)

Correct Answer: np

Q13. Which statement correctly describes the Poisson distribution?

• It models the number of successes in fixed number of independent trials with fixed p
• It approximates the binomial when n is large and p is small and has mean λ equal to its variance
• It is used only for continuous data
• Its mean is always zero

Correct Answer: It approximates the binomial when n is large and p is small and has mean λ equal to its variance

Q14. Which pharmacy application most directly uses probability to estimate patient risk?

• Calculating tablet dissolution time deterministically
• Estimating probability of adverse drug reactions in a population
• Measuring pH with a single instrument reading
• Counting tablets manually without sampling

Correct Answer: Estimating probability of adverse drug reactions in a population

Q15. Positive predictive value (PPV) of a diagnostic test depends on which factor besides sensitivity and specificity?

• The prevalence of the disease in the tested population
• The cost of the diagnostic kit
• The sample color used in assay
• The manufacturer of the test

Correct Answer: The prevalence of the disease in the tested population

Q16. To find the probability a patient has a disease given a positive test result, which concept is essential?

• Pooled variance estimation
• Bayes theorem combining prior prevalence and test characteristics
• Complement rule only
• Permutation formulas

Correct Answer: Bayes theorem combining prior prevalence and test characteristics

Q17. Which statistical approach is most relevant when estimating the probability a drug remains within specification over time?

• Survival analysis / reliability modeling of degradation
• Simple linear regression without time component
• Fisher exact test for categorical outcomes
• Chi-square goodness-of-fit for a single sample

Correct Answer: Survival analysis / reliability modeling of degradation

Q18. In acceptance sampling for tablets, the Operating Characteristic (OC) curve shows:

• The probability of rejecting every lot regardless of quality
• The probability of accepting a lot versus the lot defect rate
• The dissolution profile over time
• The mean assay value across batches

Correct Answer: The probability of accepting a lot versus the lot defect rate

Q19. How many different 3-tablet combinations can be selected from a bottle of 10 tablets when order does not matter?

• 720
• 120
• 30
• 100

Correct Answer: 120

Q20. Which best distinguishes a discrete random variable from a continuous random variable?

• Discrete takes values on a continuum; continuous takes countable values
• Discrete has a probability mass function; continuous has a probability density function
• Discrete variables have negative values only
• Continuous variables are always integers

Correct Answer: Discrete has a probability mass function; continuous has a probability density function

Q21. Measurement errors in many laboratory assays are often modeled using which distribution due to the Central Limit Theorem?

• Uniform distribution
• Normal (Gaussian) distribution
• Binomial distribution
• Poisson distribution

Correct Answer: Normal (Gaussian) distribution

Q22. The Central Limit Theorem states that as sample size increases, the sampling distribution of the sample mean approaches:

• A binomial distribution regardless of original distribution
• A normal distribution regardless of the parent population distribution (under certain conditions)
• The original skewed distribution always
• A distribution with infinite variance

Correct Answer: A normal distribution regardless of the parent population distribution (under certain conditions)

Q23. The probability that at least one tablet is defective in n independent trials with defect probability p per tablet is:

• (1 − p)^n
• 1 − (1 − p)^n
• np
• n(1 − p)

Correct Answer: 1 − (1 − p)^n

Q24. If the probability of a tablet being defective is p and a batch has n tablets, the expected number of defective tablets is:

• p/n
• n + p
• np
• n(1 − p)

Correct Answer: np

Q25. Under what conditions is the Poisson distribution a good approximation to the binomial distribution?

• n is small and p is large
• n is large and p is small, with λ = np moderate
• n and p are both exactly 0.5
• Only when p equals 1

Correct Answer: n is large and p is small, with λ = np moderate

Q26. If two events are mutually exclusive and both have non-zero probability, can they be independent?

• Yes, always independent
• No, mutually exclusive non-zero events cannot be independent
• Yes, but only if probabilities sum to 1
• Independence is unrelated to mutual exclusivity

Correct Answer: No, mutually exclusive non-zero events cannot be independent

Q27. In hypothesis testing, the Type I error (alpha) represents:

• The probability of failing to detect a true effect
• The probability of rejecting a true null hypothesis
• The expected value under the alternative hypothesis
• The probability that the null hypothesis is true

Correct Answer: The probability of rejecting a true null hypothesis

Q28. The Area Under the ROC Curve (AUC) measures:

• The cost of a diagnostic test
• The diagnostic test’s overall ability to discriminate between diseased and non-diseased states
• The prevalence of disease in a sample
• The time to result for a laboratory assay

Correct Answer: The diagnostic test’s overall ability to discriminate between diseased and non-diseased states

Q29. Probability tree diagrams are most useful for:

• Performing a single-sample t-test
• Visualizing and calculating joint and conditional probabilities for sequential events
• Calculating mean and variance only
• Designing dissolution apparatus

Correct Answer: Visualizing and calculating joint and conditional probabilities for sequential events

Q30. In pharmacoeconomics, how is expected value used when combining costs and uncertain outcomes?

• By selecting the cheapest option regardless of probabilities
• By calculating the weighted average of costs across possible outcomes using their probabilities
• By ignoring probability and using median costs only
• By multiplying the highest cost by the highest probability only

Correct Answer: By calculating the weighted average of costs across possible outcomes using their probabilities

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