Statistics and Probability Answers

In an assembly-line production of industrial robots, gearbox assemblies can be installed in one

minute each if holes have been properly drilled in the boxes and in ten minutes if the holes must be redrilled. Twenty gearboxes are in stock, 2 with improperly drilled holes. Five gearboxes must be selected from the 20 that are available for installation in the next five robots.

(a) Find the probability that all 5 gearboxes will fit properly.

(b) Find the mean, variance, and standard deviation of the time it takes to install these 5 gearboxe

A fast food chain store conducted a taste survey before marketing a new hamburger. The results of the survey showed that 75% of the people who tried this hamburger liked it. Encouraged by this result, the company decided to market the new hamburger. On a certain day, eight customers bought it for the first time. Find the probability that at least six of the eight customers will like this hamburger

If the duration of kidney disease from the onset of symptoms until kidney “failure” ranges from 1 to 5 years, the average is 2.42 years with a standard deviation of 0.89 years. One administrator of a public medical center randomly selected the medical records of 41 patients with damaged kidneys. Find the probability for :

1. the average duration is less than 3.2 years.

2. the average duration exceeds 3.2 years.

3. the average duration lies within 0.8 year of the population mean

A person’s blood glucose level and diabetes are closely related. Let x be a random variable measuring the glucose in mg per decilitre of blood. After a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean µ = 85 and standard deviation σ = 25. (Source: Diagnostic Tests with Nursing Implications, edited by S. Loeb, Springhouse Press). Note: after 50 years of age, both the mean and standard deviation tend to increase.

(a) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is less than 60?

(b) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is between 70 and 100?

(c) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is more than 125? (borderline diabetes starts at 125)

(d) Find the minimum glucose level (after a 12-hour fast) required to participate in a study reserved for the top 10% of people.

Let X and Y be independent random variables each having a geometric probability

mass function with parameter 1/2 Let Z = Y - X and M = min(X, Y) . Find the joint

p.m.f of M and Z P(M=m , Z=z) , for integers z and m > 0

Given that the variance is Q, determine the standard deviation in terms of Q.

a) 0.5Q

b) 2Q

c) Q

d) Q2

A random 5-card poker hand is dealt from a standard deck of cards. Find the probability (in terms of binomial coefficients) of getting a flush (all 5 cards being of the same suit: do not count a royal flush, which is a flush with an ace, king, queen, jack and 10).

1. State whether the following statements are True or False and also give the reason in support of your answer.

(a)Standard deviation of a random variable X may take any real value, i.e. its value lies in the interval (-infinity, infinity)

(b)If events E1,E2,...En are mutually exclusive and exhaustive then P(E1UE2U....UEn)will be greater than 1/2 but less than 1.

(c)If X is a random variable having range set {0, 1, 2, 3} then the set {0,1,2,3} then the set {x belongs to S:X(x)=0}is an event having at least one outcome of the random experiment

There are 4 black, 3 blue and 8 red balls in an urn. Three balls are selected one by one without replacement. What is the probability that:

(i)First ball drawn is black, second one is red and third one is blue

(ii)All the three balls are of the same colour

Suppose that a fair six-sided die is tossed. what is the probability of obtaining a number less or equal to three if even number have occured