Questions in Section Probability/Statistics

In a box there are nine identical marbles. Four of these marbles are red and five are not. Adam is blindfolded and selects two marbles. What is the probability that Adam selects at least one red marble?

George has some jelly beans in two bags. In one bag he has 4 yellow, 1 black, 4 orange and 3 white jelly beans. In the other bag he has 3 yellow, 4 black, 3 orange and 2 white jelly beans. If George takes one jelly bean out of each bag, what is the likelihood that both of the jelly beans are yellow?

The compressive strength of samples of cement is assumed to be normally distributed with a mean of `6000(kg)/(cm^2)` and a standard deviation of `100(kg)/(cm^2)`.

  1. What is the probability that a sample's strength is less than `5850(kg)/(cm^2)`?
  2. What compressive strength is exceed by 95% of the samples?
  3. What is the probability that at least one sample, among 4 random samples, will have a compressive strength less than `5850(kg)/(cm^2)`?
  4. If the quality control section wants to construct a `bar(x)` chart based on samples of size 6, what will be the `3sigma` control limits for the chart? If the process mean suddenly shifts to `6120(kg)/(cm^2)`, what is the probability that the shift will be detected on the next sample taken? Note that a sample mean falling above the upper control limit or below the lower control limit would signal a shift in the process mean on the `bar(x)` chart.

A manufacturing process has 95 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 3% of the components are identi ed as defective, and the components can be assumed to be independent. If the manufacturer stocks 100 components, what is the probability that all orders can be filled without reordering the components? (Give an exact solution, an approximate solution)

The number of messages sent to a bulletin board is a Poisson random variable with a mean of five messages per hour.

  1. What is the probability that more than 2 messages are received between 14:00 and 15:00?
    1. How many hits would you expect in two-hour time period?
    2. Calculate the probability of 14 or more messages in a two-hour period.
    3. Would it be unusual to receive 14 messages in a two-hour period? Explain briefly.

A hip joint replacement part is being stress-tested in a laboratory. The probability of successfully completing the test is 0.90. Eight randomly and independently chosen parts are tested.

  1. Let `X` be the number of parts completing the test successfully. Graph the probability distribution of `X` and indicate whether the distribution is skewed or symmetric.
  2. Find `P(4<=X<=5)`.
  3. Find the probability that at most 2 parts failed the test.

Suppose we have a means of generating independent fair coin flips. We want to simulate a biased coin that comes up heads with probability `p`.

  1. Give an algorithm to simulate such a coin. Assume that `p` is given in binary notation, this notation describing p may possibly be infinite if `p` is irrational.
  2. Estimate the expected number of flips of a fair coin per a simulated flip of the biased one.

Calculate the mean and variance of a Gamma random variable with parameters `alpha` and `lambda`.

Calculate the mean and variance of a Poisson random variable with parameter `lambda`.

Let `X` be a random variable with probability density distribution given by `f(x)={(x if 0<=x<=1),(1 if 1<x<=1.5),(0 quad otherwise):}`.

Find the probability density function of `Y=10X-4`.

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