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Problem Set 5 - Probability with Engineering Applications - 1998 | ECE 313, Assignments of Statistics

University of Illinois - Urbana-Champaign Statistics

Material Type: Assignment; Class: Probability with Engrg Applic; Subject: Electrical and Computer Engr; University: University of Illinois - Urbana-Champaign; Term: Fall 1998;

Typology: Assignments

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University of Illinois

Fall 1998

ECE 313:

Problem Set #5

Assigned:

September 21, 1998

Due:

September 30, 1998

Reading:

Ross, Chapter 3

1. [20 pts]

Calvin is trying his hand at gambling on a particular game in a tug-of-war tournament

on Narg. His friend, a bookie, has given him the following information on the two teams

playing: The Slugs and The Blobs.

Each team must have an anchor, on whose ample girth the teams fortunes depend. The Slugs

have only two anchors,

S

1

and

S

2

, of dierent abilities. The Blobs have three anchors:

B

1

,

their heaviest but tempermental anchor;

B

2

, their most reliable anchor;

B

3

, the worst of the

three. Furthermore:

The Blobs will not start

B

1

if

S

1

does not start. If

S

1

does start, the Blobs will start

B

1

with

probability1

=

3.

The Slugs are equally likely to start

S

1

or

S

2

,

no matter what the Blobs do

.

B

2

will anchor the Blobs with probability3

=

4if

B

1

does not anchor,

no matter what else

happens

.

The probability that the Slugs will win, given

S

1

anchors, is

m=

(

m

+ 1), where

m

is the

subscript of the Blob anchor. The probability that the Blobs will win, given

S

2

anchors, is

1

=m

, where

m

is again the subscript of the Blob anchor.

(

a)

What is the probability that

B

m

starts,

m

=1

;

2

;

3?

(

b)

What is the probability that the Slugs win, given that

S

2

and

B

m

;m

=2

;

3 anchored?

(

c)

What is the probability that the Slugs win, given that

S

2

anchored?

(

d)

What is the probability that the Slugs win?

(

e)

Given that

B

2

does not anchor, what is the probability that the Slugs win?

2.

Let

A; B

, and

C

denote the events that your mother serves respectively asparagus, broccoli,

and cauliower for dinner. From (bitter?) experience you know that these events are disjoint,

and that

P

(

A

)=0

:

2,

P

(

B

)=0

:

5,

P

(

C

)=0

:

3. Your mother makes independent decisions

(without taking your opinion into account!) about the vegetable to serve eachday. [\No, dear,

Cheetos and Oreos are not vegetables . .. "]. Over a three day period, what is the probability

that:

(

a)

she serves the same vegetable on all three days?

(

b)

she serves the same vegetable exactly twodays out of three?

(

c)

she serves dierentvegetables on the three days?

(

d)

she served broccoli on the rst day, given that all the three dierentvegetables were

served during the three days?

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University of Illinois Fall 1998

ECE 313: Problem Set

Assigned: Septemb er 21, 1998

Due: Septemb er 30, 1998

Reading: Ross, Chapter 3

1. [20 pts] Calvin is trying his hand at gambling on a particular game in a tug-of-war tournament

on Narg. His friend, a b o okie, has given him the following information on the two teams playing: The Slugs and The Blobs. Each team must have an anchor, on whose ample girth the teams fortunes dep end. The Slugs

have only two anchors, S 1 and S 2 , of di erent abilities. The Blobs have three anchors: B 1 ,

their heaviest but temp ermental anchor; B 2 , their most reliable anchor; B 3 , the worst of the

three. Furthermore:

The Blobs will not start B 1 if S 1 do es not start. If S 1 do es start, the Blobs will start B 1 with

probability 1 =3.

The Slugs are equally likely to start S 1 or S 2 , no matter what the Blobs do.

B 2 will anchor the Blobs with probability 3 = 4 if B 1 do es not anchor, no matter what else

happ ens.

The probability that the Slugs will win, given S 1 anchors, is m=(m + 1), where m is the

subscript of the Blob anchor. The probability that the Blobs will win, given S 2 anchors, is

1 =m, where m is again the subscript of the Blob anchor.

(a) What is the probability that Bm starts, m = 1 ; 2 ; 3?

(b) What is the probability that the Slugs win, given that S 2 and Bm ; m = 2 ; 3 anchored?

(c) What is the probability that the Slugs win, given that S 2 anchored?

(d) What is the probability that the Slugs win?

(e) Given that B 2 do es not anchor, what is the probability that the Slugs win?

2. Let A; B , and C denote the events that your mother serves resp ectively asparagus, bro ccoli,

and cauli ower for dinner. From (bitter?) exp erience you know that these events are disjoint,

and that P (A) = 0 :2, P (B ) = 0 :5, P (C ) = 0 :3. Your mother makes indep endent decisions

(without taking your opinion into account!) ab out the vegetable to serve each day. [\No, dear, Cheetos and Oreos are not vegetables... "]. Over a three day p erio d, what is the probability that:

(a) she serves the same vegetable on all three days?

(b) she serves the same vegetable exactly two days out of three?

(c) she serves di erent vegetables on the three days?

(d) she served bro ccoli on the rst day, given that all the three di erent vegetables were

served during the three days?

3. Supp ose that 75 passengers hold reservations for a ight from Chicago to Champaign. The

airplane has 70 seats only. Each passenger decides indep endently with probability 0.8 whether to show up for the ight.

(a) Find the probability that all passengers who show up get seats.

(b) Supp ose that 10 passengers are arriving in Chicago on a connecting ight which is late

with probability 1/4. If the connecting ight is on time, all 10 show up for the ight to Champaign; otherwise none of the 10 shows up. The remaining 65 passengers decide indep endently as b efore (and also indep endently of the fate of the connecting ight). What is the probability that all passengers who show up get a seat? Given that all the passengers who showed up got a seat, what is the (conditional) probability that the connecting ight was late?

4. Alice, Bob, and Carol play a game in which each player rolls a pair of fair dice in turn, with

Alice going rst, then Bob, and then Carol, and then Alice again, and so on. When it is Alice's turn to play, the game terminates with a win for Alice if the sum of the two dice she rolls is 8. When it is Bob's turn, the game terminates with a win for Bob if the sum of the two dice he rolls is 9. When it is Carol's turn, the game ends with a win for Carol if the sum of the dice she rolls is 10. The game continues with each player rolling the dice in turn, until one of the three players wins. Find the win probabilities for each player.

5. Supp ose that n indep endent trials of an exp eriment are p erformed, and that event A o ccurs

with probability P (A) = p on each trial. Let Pn denote the probability that A o ccurs an even

numb er of times on n trials.

(a) Show that Pn = p(1 Pn 1 ) + (1 p)Pn 1 for n 2.

(b) Use the result of part (a) to prove that Pn = 0 : 5 + 0 :5(1 2 p)n for n 1.

Notice that if A do es not o ccur at all, then it o ccurs an even numb er of times, b ecause zero

is an even numb er. This problem is taken from Ross, theoretical exercise 15, on p. 120.

6. A fair coin is tossed 10 times. What is the probability that two consecutive heads do not

o ccur among the 10 outcomes?

7. In successive rolls of a pair of fair dice, what is the probability of rolling \seven" twice (not

necessarily on consecutive rolls), b efore rolling an even numb er six times (not necessarily on consecutive rolls)? This is taken from Ross, problem 74, on p. 115.