In the transmission of digital information the probability t

In the transmission of digital information, the probability that a bit has high, moderate, or low distortion is 0.04, 0.05, and 0.91. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be independent. Let X and Y denote the number of bits with high and moderate distortion out of three transmitted, respectively. Determine the following (round all of your answers to 4 decimal places):

(a) The probability that two bits have high distortion and one has moderate distortion.

(b) The probability that all three bits have low distortion.

Solution

X denote the number of bits with high distortion

Y denote the number of bits with moderate distortion.

a)

P[two bits have high distortion and one has moderate istortion]=P[X=2,Y=1]=P[X=2]*P[Y=1] [as the aamount of distortion of each bits is assumed to be independent]

=0.04²*0.05=0.00008=0.0001 [approximated to 4 decimal]

b) P[all three have low distortion]=0.91³=0.753571=0.7536 [approximated to 4 decimals]