Question

Suppose that the security code for a particular automobile’s locks consists of 4 bytes of data transmitted via a low power radio. A byte is either sent correctly or is distorted by interference. The car locks are only opened if all 4 bytes are received correctly. Let C be the event that a byte is received correctly and F be the event that the byte is distorted in transmission. Then CCCC would mean that all four bytes were received correctly and the car locks will open, etc.

a) List the elements of the sample space for the transmission of the security code. Now let B be the event that the ith byte is received correctly.

b) List the elements of the events B1, B2, B3, and B4. Are these events disjoint?

c) List the sample points that constitute each of the following events and describe the events

in words:

i) B₁^C

ii) B₁ U B₂

iii) B₁ ? B₂

iv) B₁ ? B₂ ? B₃ ? B₄

v) B₁ ? B₂ ? B₃ ? B₄^C

vi) (B₁ U B₂ U B₃ U B₄)^C

Answer 1

(a)

In this case, the sample space will have: 2^4 = 16 possible outcomes, which are as shown below:

{ CCCC, CCCF, CCFC, CFCC, FCCC, CCFF, CFFC, FCCF, FFCC, CFCF, FCFC, CFFF, FCFF, FFCF, FFFC, FFFF }

(b)

B1 is the event where first byte is received correctly, so the possible outcomes for this event are:

{ CCCC, CCCF, CCFC, CFCC, CCFF, CFFC, CFCF, CFFF }

B2 is the event where second byte is received correctly, so the possible outcomes for this event are:

{ CCCC, CCCF, CCFC, FCCC, CCFF, FCCF, FCFC, FCFF }

B3 is the event where third byte is received correctly, so the possible outcomes for this event are:

{ CCCC, CCCF, CFCC, FCCC, FCCF, FFCC, CFCF, FFCF }

B4 is the event where fourth byte is received correctly, so the possible outcomes for this event are:

{ CCCC, CCFC, CFCC, FCCC, CFFC, FFCC, FCFC, FFFC }

No these events are not disjoint, because all of them have some common outcomes.

(c)

I could only understand the second part, all other parts are not correctly typed.

(ii) B1 U B2

This is the event in which either the first byte or the second byte or both are correct.

For this event the possbile outcomes are:

{ CCCC, CCCF, CCFC, CFCC, CCFF, CFFC, CFCF, CFFF, FCCC, FCCF, FCFC, FCFF }