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Complete the table and construct the probability histogram for a binomial random variable x with ${\displaystyle n=6\text{and}p=0.5}$.
$$\begin{array}{|cccccccc|}\hline x& 0& 1& 2& 3& 4& 5& 6\\ p(x)\\ \hline\end{array}$$

An event has a probability ${\displaystyle p=\frac{5}{8}}$. Find the complete binomial distribution for ${\displaystyle n=6}$ trials.

The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20-39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ${\displaystyle ?=500n=500}$ of young adults ages 20-39 in the United States.
Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lances

An autoclaved aerated concrete (AAC) manufacturer packs the products into boxes of 50. It is known that 0.3% of AACS produced by this manufacturer are always defective, what percentage of a randomly chosen box contains:
a) No defective at all ?
b) 3 or more defectives ?

Simplify, please: ${\displaystyle \frac{12!}{8!4!}}$

A robot has a probability to solve a puzzle of 47% if the robot tries to solve 10 puzzles. 1. What is the probability that the robot to solve less than 2 puzzles ( not including 2)? Use 4 dig 2. On average, how many puzzles will be solved by the robot:

In a recent​ poll, 45​% of survey respondents said​ that, if they only had one​ child, they would prefer the child to be a boy. Suppose you conducted a survey of 160 randomly selected students on your campus and find that 79 of them would prefer a boy. Complete parts​ (a) and​ (b) below.
Click here to view the standard normal distribution table (page 1).
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Click here to view the standard normal distribution table (page 2).
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​ (a) Use the normal approximation to the binomial to approximate the probability​ that, in a random sample of 160 ​students, at least 79 would prefer a​ boy, assuming the true percentage is 45​%.
The probability that at least 79 students would prefer a boy is 3636.
​ (Round to four decimal places as​ needed.)

The manufacture of textile assumes that the proportion of orders for late delivery of raw material is ${\displaystyle p=0.7}$. If at most 4 arrived late for a random sample of 12, reject the hypothesis that ${\displaystyle p=0.7}$ in favor of the alternative hypothesis that ${\displaystyle p<0.7}$. Use the binomial distribution.
a.) Find the probability of committing a type I error if the true proportion is 0.7.
b.) Find the probability of committing a type II error for the alternative ${\displaystyle p=0.6}$.

Students arrive for the much dreaded JEE Advanced according to a Poisson process with rate lambda. For sanitization process they must stand in a queue,each student can take different time or same time compared with some other student for sanitization. Let us denote the time taken by i-th student as ${\displaystyle X_i}$.
${\displaystyle X_i}$ are independent identically distributed random variables. We assume that ${\displaystyle X_i}$ takes integer values in range 1,2, ... , n, with probabilities ${\displaystyle p_1,p_2,\dots ,p_n}$. Find the PMF for ${\displaystyle N_t}$, the number of students in sanitization queue at time t.

Simplify the factorial expression ${\displaystyle \frac{(n+1)!}{n!}}$

During contract negotiations, a company seeks to change the number of sick days employees may take, saying that the annual “average” is 7 days of absence per employee. The union negotiators counter that the “average” employee misses only 3 days of work each year. Explain how both sides might be correct, identifying the measure of center you think each side is using and why the difference might exist.

Consider a Poisson distribution with a mean of two occurrences per time period.
Write the appropriate Poisson probability function to determine the probability of x occurrences in three time periods.

Consider a Poisson distribution with a mean of two occurrences per time period.
Compute the probability of five occurrences in two time periods.

Consider a Poisson distribution with a mean of two occurrences per time period.
Compute the probability of six occurrences in three time periods.

Consider a Poisson distribution with a mean of two occurrences per time period.
Write the appropriate Poisson probability function.

Uniform distribution is defined over the interval from 2 to 5.
Find the probability of a value between 2.9 and 3.7.

Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row of 5 seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?

Consider a Poisson distribution with a mean of two occurrences per time period.
What is the expected number of occurrences in three time periods?

Uniform distribution is defined over the interval from 2 to 5.
Find the probability of a value more than 2.6.

Consider a Poisson distribution with a mean of two occurrences per time period.
Compute the probability of two occurrences in one time period.