Question

1. A transmitter sends a binary signal (a 0 or 1), which might be corrupted and not correctly received by a receiver. The probability that a 0 is correctly received is 0.9, and the probability that a 1 is correctly received is 0.95. If each digit is a 1 or a 0 with equal probability, then given that a 1 is received, what is the probability that this is a correct reading?

2. You have a collection of 15 phones, 4 of which are broken. You inspect 9 of them. (a) What is the probability 2 are broken? (b) Given that 2 are broken, what is the probability the first is broken?

Answer 1

1)
P(1 is received)=P(1 is sent and 1 is received)+P(0 is sent and 1 is received)

=(1/2)*0.95+(1/2)*(1-0.9)=0.525

hence P(1 is sent given 1 is received)==P(1 is sent and 1 is received)/P(1 is received)

=(1/2)*0.95/0.525=0.904762

2)

a) probability 2 are broken =⁴C₂*¹¹C₇ /¹⁵C₉ =0.395604

b) P(first is broken and 2 are broken) =(4/15)*(⁴C₁*¹¹C₇ /¹⁴C₈) =0.087912

hence probability the first is broken given 2 are broken

=P(first is broken and 2 are broken)/P(2 are broken)==0.087912/0.395604=0.2222