Discrete Probability Distributions PDF

A probability distribution can be defined as a function that describes all possible values of a random variable as well as the associated probabilities. Discrete probability distribution is a type of probability distribution that shows all possible values of a discrete random variable along with t...

A probability distribution can be defined as a function that describes all possible values of a random variable as well as the associated probabilities. Discrete probability distribution is a type of probability distribution that shows all possible values of a discrete random variable along with the associated probabilities. In other words, a discrete probability distribution gives the likelihood of occurrence of each possible value of a discrete random variable. There are two conditions that a discrete probability distribution must satisfy. These are given as follows: (1) 0 ≤ 𝑃(𝑋 = 𝑥) ≤ 1 (2) σ 𝑃 𝑋 = 𝑥 = 1. Types of Discrete Probability Distributions Binomial Distribution Hypergeometric Distribution Geometric Distribution Negative Binomial Distribution Poisson Distribution Binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either success or failure. 𝑛 𝑥 𝑛−𝑥 𝑃(𝑋) = 𝑝 𝑞 𝑥 Where: 𝑃 𝑋 = probability of x successes 𝑛 = Bernoulli trials 𝑝 = Probability of successful event 𝑞 = Probability an event fails 𝑥 = number of successful events Mean: µ = 𝑛𝑝 Variance: 𝜎 2 = 𝑛𝑝(1 − 𝑝) A six-sided die is rolled 12 times. What is the probability of getting a 4 five times? 𝑛 𝑥 𝑛−𝑥 𝑃(𝑋) = 𝑝 𝑞 𝑥 A multiple-choice test contains 10 questions. Only one answer among the 4 choices to each question represents the correct answer. Find the probability that a student will answer exactly 6 questions correctly if he makes random guesses on all questions. 𝑛 𝑥 𝑛−𝑥 𝑃(𝑋) = 𝑝 𝑞 𝑥 20% of all sophomores are enrolled in Engineering and Data Analysis (EDA) course. Approximately 50 students are taken in random. Find the probability that exactly 17 students out of the 50 are taking up EDA. Determine the mean and standard deviation. The hypergeometric distribution describes the number of successes in a sequence of n trials from a finite population without replacement. 𝐾 𝑁−𝐾 𝑃 𝑥 = 𝑥 𝑛−𝑥 𝑁 𝑛 Where: 𝑁 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑 𝐾 = 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑎𝑠 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑁 − 𝐾 = 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑎𝑠 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑡𝑟𝑖𝑎𝑙𝑠 6 doctors and 19 nurses attend a small conference. All 25 names are put in a list, and 5 names are randomly picked without replacement for a raffle. What is the probability that 4 doctors out of the 5 names are picked? 𝐾 𝑁−𝐾 𝑃 𝑥 = 𝑥 𝑛−𝑥 𝑁 𝑛 Suppose a large urn contains 400 red marbles and 600 blue marbles. A random sample of 10 marbles is drawn without replacement. What is the probability that exactly 3 are red? 𝐾 𝑁−𝐾 𝑃 𝑥 = 𝑥 𝑛 − 𝑥 𝑁 𝑛 Geometric distribution is a discrete probability distribution that describes the chances of achieving success in a series of independent trials, each having two possible outcomes. 𝑃 𝑋 = 𝑥 = 𝑞 𝑥−1 𝑝 Cumulative Geometric Distribution: 𝑃 𝑋 ≤ 𝑥 = 1 − 𝑞𝑥 Mean: 1 µ= 𝑝 Variance: 2 1−𝑝 𝜎 = 2 𝑝 According to research, 25% of all cars passing along a certain road are red on weekdays. What is the probability that the 5th car will be the first red car that passes through the road on a Tuesday? QA analyst of a particular tire manufacturing company determines that 2% of all tires produced in a day are defects. A random sample of 50 tires is tested. (a) What is the probability that the 5th tire selected is a defect? (b) What is the probability that the first defect is identified among the first 20 tires? (c) How many tires would they expect to test until they find the first defective one everyday? From the latest census, about 4% of the population in a particular state works as an engineer. (a) What is the probability that the 8th person that you meet in this state is an engineer? (b) What is the probability that you find the first engineer is among the first 10 people that you meet? (c) Calculate the mean, variance, and standard deviation. If Binomial Distribution is defined as the number of successes in n Bernoulli trials, Negative Binomial Distribution is defined as the number of Bernoulli Trials until rth successes. 𝑛 − 1 𝑟 𝑛−𝑟 𝑃(𝑋) = 𝑝 𝑞 𝑟−1 Mean: 𝑟 µ= 𝑝 Variance: 2 𝑟(1 − 𝑝) 𝜎 = 𝑝2 A data analyst conducting telephone surveys must get 10 or more completed surveys before their job is finished. On each randomly dialed number, there is a 9% chance of reaching an adult who will complete the survey. (a) What is the probability that the 3rd completed survey will occur on the 10th call? (b) what is the probability that he will not finish his job after 50 calls? Suppose we inspect a sequence of tire samples from the tire company, and each tire is classified as either a defect or non- defect. Note that the probability of defective tires is 2% on a given production day. (a) What is the probability that the first defective tire will be identified on the 27th inspection? (b) What is the probability that the 3rd defective tire will be identified on the 76th inspection? An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. (a) What is the probability that the third strike comes on the seventh well that was drilled? (b) What is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells? Poisson Distribution is a discrete probability distribution that gives the probability of a number of independent events occurring in a fixed time. 𝜇 𝑥 𝑒 −𝜇 𝑃 𝑋=𝑥 = 𝑥! Cumulative Poisson Distribution: −𝜇 𝜇𝑥𝑛 𝑃 𝑋=𝑥 =𝑒 𝑥! An online shop receives an average of 12 orders per day. (a) What is the probability that the business will receive exactly 8 orders in a given day? A startup receives message, on average, 7 text messages in a 3-hour period timeframe? (a) What is the probability that the business will receive exactly 9 text messages in a 3-hour period? (b) What is the probability that the startup will receive exactly 24 text messages in 8 hours? A small business, on average, has 8 calls per hour. (a) What is the probability that the business will receive exactly 3 calls in 1 hour? (b) What is the probability that the business will receive, at most, 5 calls in one hour? (c) What is the probability that the business will receive more than 6 calls In one hour?