Bayes' theorem and simple probability distributions - One Line Questions

1. If the average number of customers arriving at a store per hour is 5, what is the probability that exactly 3 customers arrive in a given hour, assuming a Poisson process? — (5^3 * e^-5) / 3!
2. A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability that the first ball is red and the second is blue? — (5/8) * (3/7)
3. Consider two events, A and B. If P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, what is P(A|B)? — 0.5
4. If X is a normally distributed random variable with mean 50 and standard deviation 10, what is the Z-score for X = 60? — 1.0
5. Consider the following scenario: 2% of men and 0.5% of women have a certain color blindness. A person is chosen at random and found to be color blind. What is the probability that the person is male, assuming the proportion of men and women in the population is equal? — 0.75
6. If P(A) = 0.6, P(B) = 0.7, and P(A ∩ B) = 0.4, what is P(A ∪ B)? — 0.9
7. If a test for a disease is 99% accurate in detecting the disease (true positive) and 98% accurate in not detecting it when absent (true negative), and 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease? — Approximately 0.033
8. A discrete random variable X can take values 1, 2, 3 with probabilities P(X=1) = 0.3, P(X=2) = 0.4, P(X=3) = 0.3. What is the expected value of X? — 2.0
9. In a normal distribution, approximately what percentage of data falls within one standard deviation of the mean (μ ± σ)? — 68%
10. In a normal distribution, approximately what percentage of data falls within two standard deviations of the mean (μ ± 2σ)? — 95%
11. In a normal distribution, approximately what percentage of data falls within three standard deviations of the mean (μ ± 3σ)? — 99.7%
12. Which of the following is NOT a property of a probability distribution? — Probabilities can be greater than 1.
13. Which distribution is used for modeling the number of failures before the first success in a series of Bernoulli trials? — Geometric
14. Which probability distribution is often used to model the number of events occurring in a fixed interval of time or space, given a known average rate? — Poisson distribution
15. Which of the following is a common continuous probability distribution? — Normal distribution
16. A coin is tossed 10 times. What is the probability of getting exactly 5 heads, assuming the coin is fair? — C(10, 5) * (0.5)^10
17. Bayes' theorem is particularly useful for: — Updating beliefs or probabilities in light of new data.
18. What is the primary difference between a discrete and a continuous probability distribution? — Discrete distributions deal with countable outcomes; continuous distributions deal with outcomes that can take any value within a range.
19. What is the probability density function (PDF) of a standard normal distribution (μ=0, σ=1)? — f(x) = 1 / sqrt(2π) * e^(-x^2 / 2)
20. The probability density function (PDF) of an exponential distribution with rate λ is: — f(x) = λ * e^(-λx) for x >= 0
21. What is the fundamental principle behind Bayes' theorem? — It updates the probability of a hypothesis based on new evidence.
22. What is the sum of probabilities for all possible outcomes in any probability distribution? — It must be 1.
23. What is the key characteristic of a continuous probability distribution? — It describes probabilities over a range of continuous values.
24. Which of the following is a probability distribution for a discrete random variable? — Binomial Distribution
25. The variance of a binomial distribution B(n, p) is given by: — np(1-p)
26. The mean (expected value) of a binomial distribution B(n, p) is given by: — np
27. If event A and event B are independent, what is the relationship between P(A|B) and P(A)? — P(A|B) = P(A)
28. What is the formula for Bayes' theorem? — P(A|B) = P(B|A) * P(A) / P(B)
29. What is the probability mass function (PMF) for a binomial distribution B(n, p)? — P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
30. What is the probability mass function (PMF) for a Poisson distribution with parameter λ? — P(X=k) = (λ^k * e^(-λ)) / k!
31. What type of probability distribution is characterized by a fixed number of independent trials, each with two possible outcomes (success or failure), and a constant probability of success? — Binomial distribution
32. What term is used for P(A) in Bayes' theorem? — Prior probability
33. What term is used for P(B|A) in Bayes' theorem when A is the hypothesis and B is the evidence? — Likelihood
34. What term is used for P(A|B) in Bayes' theorem when A is the hypothesis and B is the evidence? — Posterior probability
35. The Normal distribution is characterized by its bell shape and is defined by two parameters: — Mean (μ) and standard deviation (σ)
36. The process of converting a normal random variable X to a standard normal random variable Z is called: — Standardization
37. In Bayes' theorem, P(B) is often calculated using the law of total probability. If we have partitions A1, A2, ..., An of the sample space, then P(B) is: — Sum of P(Ai) * P(B|Ai) for all i
38. A key property of the exponential distribution is its: — Memoryless property
39. The exponential distribution is often used to model: — The time until the next event occurs in a Poisson process.
40. What is the parameter lambda (λ) in a Poisson distribution? — The average number of events in the given interval.
41. In the context of Bayes' theorem, what does P(B|A) represent? — The probability of event A occurring given that event B has occurred.
42. The memoryless property of the exponential distribution means: — The probability of an event occurring in the future does not depend on how much time has already passed.
43. What is the parameter 'rate' (often denoted by λ) in an exponential distribution? — The inverse of the mean time between events.
44. In a binomial distribution B(n, p), what does 'n' represent? — The total number of trials.
45. What is the formula for standardizing a normal random variable X with mean μ and standard deviation σ? — Z = (X - μ) / σ
46. What is the mean of an exponential distribution with rate λ? — 1/λ
47. What is the variance of an exponential distribution with rate λ? — 1/λ^2
48. For a Poisson distribution, the mean is equal to: — λ
49. For a Poisson distribution, the variance is equal to: — λ
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