Probability Exercises: Binomial, Poisson, and Normal Distributions

Posted on Feb 4, 2025 in Mathematics

Binomial Distribution

  1. What is the probability of getting three prime numbers in five rolls of a die?
  2. When flipping a fair coin, what is the probability of getting at least four heads in five flips?
  3. The state of Pennsylvania has a daily lottery. Every night, a three-digit number is chosen. What is the probability of obtaining a number less than 100 more than five times in a week?
  4. You are hunting the whale Moby Dick. Every day, you check your ship with a boat of harpooners. There is a probability of 2/3 that one of these boats will sink in a day. You plan to hunt Moby Dick for 4 days. What is the probability of losing three or more boats?
  5. Suppose you attend an exam with 100 true-false questions. You know nothing about the subject of the examination and will answer the questions by guessing. What is the chance of hitting at least 60 questions (use approximation)?
  6. How often do we flip a coin so that the probability of appearing at least two faces is greater than 1/2?
  7. Suppose that 10% of the population is left-handed. If three people are chosen at random, what is the probability that at least one is left-handed?
  8. What is the probability that two of the next three presidents of Brazil were born on a Sunday?
  9. Suppose that two-fifths of the population have blood type 0+. If six people are randomly chosen, what is the probability that four of them have 0+ blood?
  10. Suppose that 45% of the Almeidas in the world are women. If three Almeidas are chosen randomly, what is the probability that at least two are women?

Poisson Distribution

  1. Let X be a random variable representing the number of times the word “platypus” is pronounced in a day. Assuming that X has a Poisson distribution with parameter m = 1/2, what is Pr(X > 1)?
  2. If X is a Poisson random variable with parameter m = 10, what is Pr(X ≤ 3 ≤ 1)?
  3. Let X be a Poisson random variable with parameter m = 3, representing the number of people who use a dictionary in a library on any given day. What is the value of P(X ≤ 4)?

Normal Distribution

  1. Suppose that the rainfall in a city has a normal distribution with a mean of 40 and a standard deviation of 5. What is the probability that the city has less than 33 inches of rain next year? What is the probability that the city has more than 38 inches of rain?
  2. Suppose that a student’s score for college is a random variable selected from a normal distribution with a mean of 550 and a variance of 900. If admission to some college requires a score of 575, what is the probability of being admitted? And if the minimum score is 540?
  3. Suppose you are measuring the speed of light. The results of your measurements are given by a normal random variable whose average is the true value and whose standard deviation is 5 x 109 centimeters per second. What is the probability that your measurement is less than 2 x 109 centimeters per second from the real value?

In Exercises 17 to 21, X is a normal random variable with parameters μ and σ2. With the aid of a normal table, calculate:

  1. If μ = 0 and σ2 = 100, what is P(5 < X < 10)?
  2. If μ = -3 and σ2 = 9, and P(X < a) = 0.6, what is a?
  3. If μ = 0 and P(X < 5) = 0.8, what is σ2?
  4. If μ = 73 and σ2 = 81, what is P(|X| > 100)?
  5. If μ = 25 and σ2 = 100, what is P(X = 25)?

Statistics

Professor Otávio Henrique dos Santos Figueiredo

Answers:

  1. 5/16
  2. 3/16
  3. 1/156250
  4. 16/27
  5. 0.02872
  6. 4
  7. 0.271
  8. 18/343
  9. 432/3125
  10. 0.425
  11. 0.0902
  12. 0.01029
  13. 0.81526
  14. 0.08; 0.66
  15. 0.20; 0.63
  16. 0.31
  17. 0.15
  18. -2.25
  19. 35.43
  20. 0
  21. 0