A hospital reports an average of $365$ patients admitted for treatment each year. Determine the mean number of patients admitted per day.

In a small city, an average of $15$ cars are reported stolen each month. The probability distribution follows the Poisson model. What is the probability that on a given day, more than one car is stolen?

Find the mean, variance, and standard deviation for the following probability distribution.

With an average of $2$ volcanic eruptions per year in a certain area, use the Poisson distribution to find the expected number of years with exactly $3$ eruptions in a $100$-year period.

A discrete probability distribution has five outcomes with probabilities $0.12,0.18,0.25,0.05,p$. Determine the missing probability $p$ that makes the distribution valid.

A customer service agent successfully resolves a caller’s issue on the first attempt $72\%$ of the time. Find the probability that:

i. The first successful resolution occurs on the third call,

ii. The first successful resolution occurs on or before the second call, and

iii. The agent does not successfully resolve any of the first three calls.

In a certain city, the daily water usage for households with two people has a mean of $180$ liters and a standard deviation of $20$ liters. For households with three people, the mean is $250$ liters and the standard deviation is $30$ liters. If one household of each type is selected at random and their usages are independent, what is the mean and standard deviation of the total water usage?

A manufacturer produces bolts with a 2% defect rate. If 50 bolts are inspected, what is the probability that exactly 1 is defective?

A marketing research firm asked a sample of $1000$ consumers what influenced their decision most when purchasing a new smartphone. The table below summarizes the responses:

Does this table represent a probability distribution? Why or why not?

A history quiz contains three multiple-choice questions, each with four possible answer choices (a, b, c, d), only one of which is correct. Suppose a student randomly guesses on all three questions.

Beginning with WWC (two wrong answers followed by one correct answer), make a list of different possible arrangements of two wrong answers (W) and one correct answer (C). What is the probability of getting exactly one correct answer when three guesses are made?

A clinical trial for a new vaccine shows a $70\%$ chance of success. Suppose the vaccine is given to three people. Construct the binomial distribution for this situation.

A hospital records a total of $180$ patient admissions over a $30$-day month. Assuming admissions occur independently and at a constant average rate, what is the probability that at most $3$ patients are admitted on a randomly selected day?

A multiple-choice quiz has $10$ questions, each with $4$ answer choices and only one correct answer. If a student guesses on each question, what is the probability that the student gets more than $6$ questions correct? Is this event unusual? Explain your reasoning.

$62\%$ of employees prefer email as their primary method of workplace communication. You randomly select $10$ employees and ask whether they prefer email as their primary communication method.

The random variable $x$ represents the number of employees who prefer email.

i. What is the most appropriate description of the histogram's shape?

ii. Which value of $x$ would be considered unusual?

iii. Why is this value considered unusual?

A call center receives an average of $3.2$ calls per hour. Assuming the number of calls follows a Poisson distribution, what is the probability that in a randomly selected hour, the call center receives exactly $3$ calls?