Question

The probability that a delivered message contains one or more errors is 0.1. If it contains errors, they are all corrected with probability 0.8 before the message is received.

a) Determine the probability that the message is received intact (error free). hint use Law of total probability

b) If the message received is intact, what is the probability that it originally contained no errors? hint use baye's theorem

c) If 6 messages are sent, what is the probability that at least 5 are received error free?

Answer 1

We have given the following probabilities,

Define A=delivered message contains error

B=error gets corrected before message received

So P(A)=0.1, P(B|A)=0.8

To find the below probabilities,

a) P(B)=P(message received error free)

=P(error free|A)*P(A)+P(error free|A')*P(A')

=0.8*0.1+1*(1-0.1)=0.98

b)

P(Originally no error| intact message)

=P(intact message|originally no error)*P(originally no error)/[P(

intact message|originally no error)*P(originally no error)+

P(intact message|originally error)*P(originally error)]

=1*(1-0.1)/[1*(1-0.1)+0.8*0.1]

=0.9/0.98=0.9184

C) let X be the random variable denoting number of error free message among 6 messages.

p=probability of error free message=0.98

To find,

P(X>=5)

=P(X=5)+P(X=6)

=6*(0.98)⁵(1-0.98)+(0.98)⁶

=0.9943