Fundamentals of Probability - Probability Distributions (Discrete, Continuous) | AI tutor The No.1 Homework Finishing Free App

'In a certain country, it is said that the distribution of blood types among its citizens is O type 30%, A type 35%, B type 25%, and AB type 10%. Now, when randomly selecting 400 people, find the probability that the number of AB type individuals is between 37 and 49.'

'Find the probability distribution, expected value E(X), variance V(X), and standard deviation σ(X) of the number of heads X when throwing two coins.'

'Translate the given text into multiple languages.'

'A random variable X can take any value in the interval [0,10], and its probability density function is given by f(x)=kx(10-x) (where k is a constant). In this case, k='

'Explain Type I error and Type II error.'

'Distribution of sample means\nWhen a random sample of size n is taken from a population with a population mean m and population standard deviation σ, the sample mean X̅ approximately follows a normal distribution N(m, σ^2/n).'

'When a random variable X follows a normal distribution N(m, σ^2), find P(|X-m| ≥ σ/4). Round the answer to the fourth decimal place.'

'Prove that the number of times event A occurs, X, in n trials where the probability of event A occurring is p, follows a binomial distribution. Also, calculate the mean, variance, and standard deviation.'

'When the proportion of elements in a population with a certain characteristic A (population proportion) is p, what distribution can the sample proportion R be considered to follow?'

"Let's deepen our understanding of the binomial distribution."

'For a binomial distribution B(n, p) with a random variable X, show the approximate normal distribution formula that X follows when n is large. Let q = 1 - p.'

'When the random variable X follows the normal distribution N(m, σ^2), show the transformation formula to obtain the standard normal distribution Z.'

'(1) When the random variable Z follows the standard normal distribution N(0,1), find the probability P(-1.98 ≤ Z ≤ -0.5).\n(2) When the random variable X follows the normal distribution N(30,4²), find the probability P(22 ≤ X ≤ 32).'

'Using the normal distribution table, solve the following problem: Find the probability when u = 0.4 and z = 0.73.'

'The defect rate of a certain product is approximately 7%. How many products need to be sampled to ensure that the width of the confidence interval for the defect rate, with a confidence level of 95%, is less than or equal to 4% and 2%, respectively?'

'Please calculate the percentage (%) of deviation values y greater than 65. Given that the mean value of y is 50 and the standard deviation is 10, let z=(y-50)/10, then z follows the standard normal distribution N(0,1).'

'Seat allocation in elections'

'Since the sample size n is 400, the 95% confidence interval for the population mean m is 51.0-1.96*(9.5 / sqrt(400)) <= m <= 51.0 + 1.96*(9.5 / sqrt(400)), therefore 50.069 <= m <= 51.931'

'Probability distribution, binomial distribution'

'When two dice are rolled simultaneously, find the probability distribution of the random variable X. If the two dice have the same number, that number will be considered as X.'

'When the probability density function f(x) of the random variable X is given by the following equation, find the specified probabilities.'

"Let's review the binomial distribution!"

'Prepare 3 cards with the number 1, 3 cards with the number 2, and 3 cards with the number 3, totaling 9 cards. When randomly selecting 3 cards from these, what is the probability that the sum of the numbers on the cards is a multiple of 3?'

"Please demonstrate that events A and B become mutually exclusive when event B is changed to 'ball (3) appears'."

"The event of 'at least 2 girls standing consecutively' is the complementary event to 'girls not being next to each other'. Girls not being next to each other occurs when 3 girls are standing between 10 boys. The total number of circular permutations of 10 boys is (10-1)!=9! (ways). For each of these cases, there are 10P3 ways for 3 girls to stand in 3 out of 10 positions. Therefore, the number of ways for girls not being next to each other is 9! × 10P3 (ways). Hence, the required probability is P(̅A) = 1 - P(A) = 1 - (9!×10P3/12!) = 1 - 10×9×8/12×11×10 = 5/11."

"Let's assume there is a biased die with probabilities of rolling 1, 2, 3, 4, 5, 6 as 1/6, 1/6, 1/4, 1/4, 1/12, 1/12 respectively. Calculate the probability of getting a sum of 6 when rolling the die three times in a row. [Tokyo Denki University]"

'Translate the given text into multiple languages.'

'526'

"Suppose 4 people a, b, c, d from probability distribution A are divided into two groups {a, b} and {c, d}, and each person must choose one person from the other group with equal probability. Assume that each person's choice is independent. Let X be the number of pairs who choose each other. Find: (1) The distribution of the random variable X. (2) Find the expected value of X."

'Basic Example 62 Binomial Distribution Mean and Variance\nFrom a bag containing 6 red balls and 4 white balls, one ball is drawn and then replaced. Repeat this process 6 times, let X be the number of times a red ball appears, calculate the expected value E(X), variance V(X), and standard deviation σ(X) of X.'

'Assuming the grades follow a normal distribution with m = 62, σ = 20, what grade will a student who scored 85 receive?'

'Item: Statistical Inference\nTopic: Probability Distribution\nQuestion Number: 6\nQuestion Content: Explain about probability distribution.'

'Consider all students in a high school as the population, with a population proportion of students who never read books being 0.5 and a random sample size of 100. When the random variable X follows a binomial distribution B(100, 0.5), calculate the mean (expected value) and standard deviation of X. Also, assuming a population proportion of 0.5 for students who never read books, show the probability when X approximately follows a normal distribution.'

'Roll two dice simultaneously, let X be the minimum of the two results, determine the probability distribution of X. Also, calculate P(X ≤ 3).'

'The probability of rolling a 1 with one throw of a die is 1/6.'

'Let X be a random variable following a binomial distribution with mean 6 and variance 2. Let Pk be the probability when X = k. Find the value of P4/P3.'

'Find the value of the positive constant a when the probability density function of random variable X is given by f(x).'

'Basic Example 68 Use of the Normal Distribution\nLet X be the height of male students in a certain high school, following a normal distribution with mean 170.9 cm and standard deviation 5.4 cm. Answer the following questions. Round to one decimal place.\n(1) What percentage of students have a height of 175 cm or higher?\n(2) What height is required to be taller than approximately 4% of students?'

'If a new drug was used on 400 patients and 8 of them experienced side effects, can we say that the rate of side effects of this new drug is not 4% assuming the rate of side effects for the drug that has been used traditionally is 4%? Perform a hypothesis test with a significance level of 5%. Also, how about at a significance level of 1%? Assume that the 400 patients were randomly sampled.'

'When the random variable X follows a normal distribution N(15,3^2), find the following probabilities:\n(1) P(X ≤ 18)\n(2) P(6 ≤ X ≤ 21)'

'At university A, 64% of all students supported issue X. At another university B, out of 400 randomly selected students, 274 students supported X. Can it be said that there is a difference in the support rate for X between students at B and students at A? Test at a significance level of 5%.'

'Explain how to find a random variable Z that follows a standard normal distribution.'

'Assuming the distribution of exam scores follows a normal distribution with a mean of 58.4 and a standard deviation of 25. Please calculate the probability that the average score of 100 randomly selected individuals taking the test is at least 62 points.'

'Calculate probability distribution'