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Probability Homework Solutions for ISyE 6739 – Summer 2005, Assignments of Data Analysis & Statistical Methods

Georgia Institute of Technology - Main CampusData Analysis & Statistical Methods

Solutions for homework problems related to probability theory, covering topics such as combinations, independent events, and conditional probability. Students can use this document to check their understanding of these concepts and to prepare for exams.

Typology: Assignments

Pre 2010

Uploaded on 08/05/2009

koofers-user-y6z
koofers-user-y6z 🇺🇸

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ISyE 6739 – Summer 2005
Homework #2 (Covers Modules 6–9) — Due Thursday, June 9
Just do 10 or 11 of these problems,
1. A bridge hand contains 13 cards from a standard deck. Find the probability that
a bridge hand will contain
(a) Exactly 2 aces.
(b) At least 2 aces.
(c) 8 spades.
(d) 8 cards of the same suit.
2. A die is thrown 7 times. Find
(a) Pr(‘6’ comes up at least once).
(b) Pr(each face appears at least once).
3. Write a computer program in your favorite language to calculate combinations.
Demonstrate your program on C100,50.
4. Twenty items (12 bad, 8 good) are inspected. If the items are chosen at random,
what’s the probability that
(a) The first two are bad?
(b) The first two are good?
(c) One of each in the first two?
5. A box contains 4 bad and 6 good tubes. Two are drawn out. One is tested and
found to be good. What’s the probability that the other is good?
6. In a class there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How
many sophomore girls must be present if gender and class are to be independent
when a student is selected at random?
7. If Aand Bare independent, Pr(A) = 0.4, and Pr(A∪B) = 0.6, find Pr(B).
8. Suppose Pr(A) = 0.4, Pr(A∪B) = 0.7, and Pr(B) = x.
(a) For what choice of xare Aand Bdisjoint?
(b) Independent?
pf2

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ISyE 6739 – Summer 2005

Homework #2 (Covers Modules 6–9) — Due Thursday, June 9 Just do 10 or 11 of these problems,

  1. A bridge hand contains 13 cards from a standard deck. Find the probability that a bridge hand will contain (a) Exactly 2 aces. (b) At least 2 aces. (c) 8 spades. (d) 8 cards of the same suit.
  2. A die is thrown 7 times. Find (a) Pr(‘6’ comes up at least once). (b) Pr(each face appears at least once).
  3. Write a computer program in your favorite language to calculate combinations. Demonstrate your program on C 100 , 50.
  4. Twenty items (12 bad, 8 good) are inspected. If the items are chosen at random, what’s the probability that (a) The first two are bad? (b) The first two are good? (c) One of each in the first two?
  5. A box contains 4 bad and 6 good tubes. Two are drawn out. One is tested and found to be good. What’s the probability that the other is good?
  6. In a class there are 4 freshman boys, 6 freshman girls, and 6 sophomore boys. How many sophomore girls must be present if gender and class are to be independent when a student is selected at random?
  7. If A and B are independent, Pr(A) = 0.4, and Pr(A ∪ B) = 0.6, find Pr(B).
  8. Suppose Pr(A) = 0.4, Pr(A ∪ B) = 0.7, and Pr(B) = x. (a) For what choice of x are A and B disjoint? (b) Independent?
  1. Prove: If Pr(A|B) > Pr(A), then Pr(B|A) > Pr(B).
  2. The probability of scoring a basket is p. Joe shoots first. If he misses, Fred gets to shoot. They shoot the ball back and forth until somebody scores. What’s the probability that Joe wins? Graph this as a function of p.
  3. Consider two boxes. The first box contains one black marble and one white marble. The second box contains two blacks and one white. A box is selected at random and a marble is drawn at random from the selected box. (a) Find Pr(the marble is black). (b) What is the probability that the marble was selected from the first box given that the marble is white?
  4. A gambler has in his pocket a fair coin and a two-headed coin. (a) He selects one at random, and when he flips it, it shows heads. What’s the probability that the coin is fair? (b) He flips the same coin and it again shows heads. Same question. (c) He flips it a third time and it shows tails. Same question.
  5. (A tough problem!) 3 prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other 2 are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information, since he already knows that at least one will go free. The jailer refuses by arguing that if A knew, A’s probability of being executed would rise from 1/3 to 1/2 (i.e., there would only be 2 prisoners left). Whose reasoning is correct and why?