SOLUTION: 5) Alice is trying to send a message to Bob. Eve is an eavesdropper who randomly corrupts words sent by Alice. Because of Eve, any word sent by Alice reaches Bob correctly with a p

Question 1020882: 5) Alice is trying to send a message to Bob. Eve is an eavesdropper who randomly corrupts words sent by Alice. Because of Eve, any word sent by Alice reaches Bob correctly with a probability, p, independent of other words. Alice sends messages to Bob such that each message is 2 words long. They also exchange some code language so that if any 1 of the two words sent by Alice is received correctly by Bob, then Bob can understand the message correctly. If a message is not received correctly by Bob, he indicates that to Alice and she re-sends the message. This is continued until Bob receives the message correctly.
a)What is the probability (in terms of p) that a message sent by Alice is received incorrectly by Bob?
b)Call your solution of Problem # 5a) as q. What should be the minimum value of q if Alice wants to give Bob a 98% guarantee that she will deliver a message correctly to Bob in 2 attempts or less?
c) What should be the minimum value of p so that you can achieve the value of q you obtained in Problem # 5b)?
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Question:
5) Alice is trying to send a message to Bob. Eve is an eavesdropper who randomly corrupts words sent by Alice. Because of Eve, any word sent by Alice reaches Bob correctly with a probability, p, independent of other words. Alice sends messages to Bob such that each message is 2 words long. They also exchange some code language so that if any 1 of the two words sent by Alice is received correctly by Bob, then Bob can understand the message correctly. If a message is not received correctly by Bob, he indicates that to Alice and she re-sends the message. This is continued until Bob receives the message correctly.
a)What is the probability (in terms of p) that a message sent by Alice is received incorrectly by Bob?
b)Call your solution of Problem # 5a) as q. What should be the minimum value of q if Alice wants to give Bob a 98% guarantee that she will deliver a message correctly to Bob in 2 attempts or less?
c) What should be the minimum value of p so that you can achieve the value of q you obtained in Problem # 5b)?

Solution:
(a)
n=2
probability of correct word = p
probability of incorrect word = (1-p)
probability of incorrect (2-word) message, q =

(b)
A guarantee is a guarantee (100%).
I suppose what is meant is what is the value of q such that 98% of the time, in the long run, a message will be received correctly in 2 attempts or less.
We need q^2 (both attempts fail) <=2%.
so q<=sqrt(1-98%)=0.14142 (approximately).
so maximum value of q so that the message will be received correctly in 2 attempts or less is sqrt(.02)=sqrt(2/100)=sqrt(2)/10.

(c)The minimum value of p is such that
(1-p)^2<=sqrt(2)/10
Solve for p:
1-p<=sqrt(sqrt(2)/10)
p>=1-sqrt(sqrt(2)/10)
=1-0.3761
=0.6239 (approximately)

Check:
((1-0.6239)^2)=0.02, ok.
Also,
in order that you get answers as fast as possible, I suggest you post new questions instead of in the messages. You are welcome to alert me to the question in your message if you so wish.
RELATED QUESTIONS
Alice and Bob play three chess games. Alice is 10 times more likely to win a game than... (answered by ikleyn,greenestamps)

Alice borrowed a bicycle from her friend, Bob, and wanted to return it. She rode the... (answered by ankor@dixie-net.com)

Hi A sum of money is split between Bob Ted and Alice in the ratio 6:2:5 respectively.If... (answered by josgarithmetic,greenestamps,ikleyn)

At present, Bob is twice as old as Alice. In 20 years time, the ratio of Alice to Bob�s... (answered by Fombitz)

Alice and Bob started to walk towards each other�s home and then back to theirs, with... (answered by ikleyn)

Alice and Bob each have a certain amount of money. If Alice receives dollars from Bob,... (answered by ankor@dixie-net.com)

Bob and Alice are playing Texas Hold 'Em. Bob is dealt cards 1 and 3 while Alice is dealt (answered by edjones)

At 5:12:40 PM Bob starts downloading a 952MB media file at 16.5Mbps. At 5:15:02 PM... (answered by Theo)

Alice is twice as old as Judy, who is younger than Alex by 3 years. If the sum of their... (answered by College Student)