An encryption-decryption system consists of three

Answer

Step 1 of 2

(a)

An encryption-decryption system consists of three elements: encode, transmit, and decode.

A faulty encode occurs in  of the messages processed, transmission errors occur in  of the messages, and a decode error occurs in  of the messages.

Assume the errors are independent.

we are asked to find the probability of a completely defect-free message.

Let  be the event of faulty encode of the messages processed.

Let  be the event of transmission errors occurs in the messages.

Let  be the event of decode errors occurs in the messages.

The probability of messages processed has a faulty encode is,

The probability of messages processed does not have a faulty encode is,

……….(1)

The probability of messages processed has transmission errors is,

The probability of messages processed does not have transmission errors is,

………(2)

The probability of messages processed has a decode errors is,

The probability of messages processed does not have a decode errors is,

………(3)

Since the errors are independent.

Hence we can apply the independence rule of probability which is,

If two events  and  are independent, then we can write,

……….(4)

Similarly, for multiple events  are independent if and only if for any subset of these events,

……(5)

Then by equation (5), we can write the probability of a completely defect-free message,

Using equation (1), (2) and (3),

Hence the probability of a completely defect-free message is