Math227 Fall2007 Homework#04

  1. Four possible probability distribution functions for a discrete random variable are shown in the table. For each, tell if the function is suitable for a probability distribution function. If not, tell why not. If yes, then find the mean and standard deviation of X.

A. B. C. D.

X / P(X) / X / P(X) / X / P(X) / X / P(X)
1
2
3
4
5
6 / 0.1
0.2
0.2
0.1
0.1
0.2 / 1
2
3
4
5
6 / 0.1
0.3
0.2
0.1
0.1
0.2 / 1
2
3
4
5
6 / 0.1
0.4
-0.2
0.2
0.1
0.2 / 1
2
3
4
5
6 / 0.1
0.3
1.1
0.3
0.1
0.2
  1. A psychologist thanks that listening to Mozart's music helps people think. She gives subjects a set of puzzles, and measures how many they solve in five minutes while listening to Mozart music. From data on many people, the psychologist gets the probability distribution function:

Puzzles Solved / 0 1 2 3 4
Probability / 0.1 0.1 0.4 0.3 0.1
  1. What is the probability a subject solves more than one puzzle?
  2. What is the probability a subject solves at least one puzzle?
  3. What is the probability a subject solves fewer than 3 puzzle?
  4. What is the probability a subject solves 2 or more puzzle?
  5. What is the probability a subject solves at the most 2 puzzle?
  6. What is the expected number of puzzles the subject will solve?
  7. What is the probability the subject solves at least one puzzle but less than four puzzles?
  1. A candidate for public office is thought to have support from 60% of the voters. If we select a random sample of six voters and form a table to find the probability distribution and then find the probability that
  1. there will be exactly four who support the candidate?
  2. there will be no one support the candidate?
  3. there will be at least four who support the candidate?
  4. there will be five or more who support the candidate?
  5. Find the mean # of people who will support the candidate.
  6. Find the variance # of people who will support the candidate.
  7. Find the standard deviation # of people who will support the candidate.
  1. A fellow student claims that 10% of the M&M candies are red. You take a random sample of 10 candies. Let X be the number of red M&M candies in the sample of size 10.
  1. Find the probability distribution of X.
  2. What is the expected number of red candies in your sample?
  3. Standard deviation for the number of red candies in your sample?
  4. What is the probability you will find exactly two red candies?
  5. What is the probability that you will find more than one red candies?
  1. The probability of randomly selecting the correct response on a multiple choice question with five choices is 0.20 (assuming zero knowledge). Suppose an exam consists of 25 multiple choice questions, each with five choices.
  1. How many correct responses would you expect a student to pick by randomly selecting answers?
  2. A student randomly answered to all 25 and had 10 questions correct. Is this unusual? Please explain in details.
  1. A physician knows the probability that a patient receiving will recover from a certain disease is 0.25 when treated with the current medication. To test the effectiveness of a new medication, he gives it to 12 patients having this disease and he decides, before conducting the study, to go to the new medication only if at least five of these patients recover.
  2. Write an expression for the probability that he goes to the new medication even though it is only equally effective as the current medication.
  3. Write an expression for the probability that he fails to go to the new medication event though the recovery rate for it is 0.50

7-Depakote is a medication whose purpose is to reduce the pain associated with migraine headaches. In clinical trials of Depakote, 2% of the patients in the study experienced weight gain as a side effect. Suppose a random sample of 30 Depakote users is obtained. Find the following probabilities:

  1. Exactly 3 experienced weight gain as a side effect
  2. 3 or fewer experienced weight gain as a side effect
  3. 4 or more experienced weight gain as a side effect.
  4. Between 1 and 4 patients, inclusive, experienced weight gain as a side effect.

9.It is estimated that 75% of a grapefruit crop is good; the other 25% have rotten centers that cannot be detected unless the grapefruit are cut open. The grapefruit are sold in sacks of 10. Let x be the number of good grapefruit in a sack.

a)Make a histogram of the probability distribution of x.

b)What is the probability of getting no more than one bad grapefruit in a sack? What is the

c)probability of getting at least one good grapefruit in a sack?

d)What is the expected number of good grapefruit in a sack?

e)What is the standard deviation of the r probability distribution?

f)What is the probability of getting at least one bad grapefruit in a sack

g)What is the probability of getting at most 3 bad grapefruit in a sack

h)What is the probability of getting more than one bad grapefruit in a sack.

10. According to Harper’s Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 20 federal inmates is selected.

i)What is the probability that 15 or more are serving time for drug dealing?

j)What is the probability that fewer than 10 are serving time for drug dealing?

k)What is the expected number of inmates serving time for drug dealing?

l)What is the standard deviation number of inmates serving time for drug dealing?