2021/8/17 HW2 – Intro to Probability

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Intro to Probability – Homework Assignment 2

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 An urn contains 3 balls: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 ball from the urn, replacing it in the box, and drawing a second ball. Describe the sample space of this experiment and for the experiment obtained not replacing the first ball in the urn.

Problem 2 In an experiment, a die is rolled continually until a 5 appears, at which point the experiment stops. What is the sample space of this experiment? Let denote the event that rolls are necessary to conclude the experiment. What outcomes are contained in

? What does the event represent?

Problem 3 Two dice are rolled. Using that is the event that the sum of the dice is odd, the event that at least one of the dice lands on 1, and the event that the sum is 5, describe the events , , , , and .

Problem 4 A hospital administrator codes incoming patients suffering from heart disease according to whether they have insurance (code 1 if they do and 0 if they do not) and based on their condition, which can be rated as good G, fair F, or serious S. Consider an experiment that consists of producing codes for such a patient.

a. Give the sample space of this experiment.

b. Let be the event that the patient is in serious condition and list all outcomes in .

c. Let be the event that the patient is uninsured and list all outcomes in .

d. List the outcomes in .

Problem 5

En n

En (⋃ ∞ n=1 En)

c

O O1 S5

O ∩ O1 O ∪ O1 O1 ∩ S5 O ∩ O c

1 O ∩ O1 ∩ S5

S S

U U

S c ∪ U

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2021/8/17 HW2 – Intro to Probability

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Suppose that two events , are such that and satisfy and . What are the probabilities that

a. either or occurs?

b. occurs but does not?

c. both and occur?

What happens if you don’t assume that ?

Problem 6 At a summer camp three activities are offered: tennis, swimming, and archery. The courses are open to the 120 summer camp participants. There are 31, 24, and 17 in these activities, respectively. There are 4 participants in all activities, 11 in tennis and swimming, 9 in tennis and archery, 7 in swimming and archery.

a. If a participant is chosen at random, what is the probability that he or she is taking part exactly one activity?

b. If a participant is chosen at random, what is the probability that he or she is taking part in no activity?

c. If two participants are chosen at random, what is the probability that they are both taking part in exactly one activity?

Problem 7 Consider three fair dice with the following digits on their faces. The blue die has 2 faces each with 1, 5, and 9 on them. The red die shows the digits 3, 4, and 8 twice each and the green one 2,6, 7 twice each. Two players choose a die each and then roll it. The winner is the player who rolls the highest number. If you had the opportunity to choose whether to pick your die first, would you do it or rather go second?

Problem 8 Let E, F, and G be three events. Use them to describe the following events:

a. Of the three only occurs.

b. Both E and G occur , but F does not.

c. At least one event occurs of the three.

d. At least two of the events occur.

e. None of the events occur.

f. All occur.

E F E ∩ F = ∅ P (E) = 0.3 P (F ) = 0.5

E F

E F

E F

E ∩ F = ∅

F

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2021/8/17 HW2 – Intro to Probability

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g. Exactly two events occur.

h. At most three of the events occur.

i. At most one of them occurs.

Problem 9 If and , show that . In general, prove Bonferroni’s inequality, i.e., that

Problem 10 An urn contains yellow and black balls. They are withdrawn one at a time until a total of yellow ones one are obtained. Find the probability that this happens on drawing number for any .

P (E) = 0.9 P (F ) = 0.8 P (E ∩ F ) ≥ 0.7

P (E ∩ F ) ≥ P (E) + P (F ) − 1

n m y ≤ n

k k ≤ m + n

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