Problem 1

a.
i.   P(circular) =  (1 - P(square)) = 1 - .6^2 = 0.64%
ii.  P(square) = .6^2 = 0.36
iii. P(blue) = .5^2 = 0.25
iv.  P(red) = 1 - P(blue) = .75

b. 
i. P(blue) still = 0.25
ii. P(red) still = .75

c. also independant
i.  P(square) still = 0.36
ii. P(circular) still = 0.64%

d. T = tested positive
   M = mutant
   
   P(T | M)  = 0.9
   P(T | ~M) = 0.005
   P(M) = 0.001

   We are looking for P(M | T):
   P(T | M)  = P(T ^ M)  / P(M)
   P(T | ~M) = P(T ^ ~M) / P(~M)

   P(M | T) = P(T ^ M) / P(T)

   P(T ^ M) = P(T | M) * P(M) = 0.9 * 0.001 = 0.0009
   P(T) = P(T | M) + P(T | ~M) = 0.9 + 0.005 = 0.905

   P(M | T) = 0.0009 / 0.905 = 0.0994 / 100 (!!!)

Probem 2

a. Each character would need 3 (log2(8)) bits.
   The phrase would need 84 (28 * 3) bits.

b. Theorecal min number of bits required
   %% MatLab
   M = [35.71 14.29 10.71 17.86 3.57 3.57 3.57 10.71]./100
   P = M .* log2(1./M)
   n = sum(P)
   Thus n = 2.5808

c. P = [0.5305 0.4011 0.3452 0.4439 0.1716 0.1716 0.1716 0.3452]

d. 
   {C="" 0.3571} {F="" 0.1789} {D="" 0.1429} {E="" 0.1071} {R="" 0.1071} {G="" 0.0357} {A="" 0.0357} {C+="" 0.0357}
   {C="" 0.3571} {F="" 0.1789} {D="" 0.1429} {E="" 0.1071} {R="" 0.1071} {G="" 0.0357} {A=1 C+=0 0.0714}
   {C="" 0.3571} {F="" 0.1789} {D="" 0.1429} {E="" 0.1071} {R="" 0.1071} {G=1 A=01 C+=00 0.1071}
   {C="" 0.3571} {F="" 0.1789} {D="" 0.1429} {E="" 0.1071} {R=1 G=01 A=001 C+=000 0.2142}
   {C="" 0.3571} {F="" 0.1789} {D="" 0.1429} {E=1 R=01 G=001 A=0001 C+=0000 0.3213}
   {C="" 0.3571} {F="" 0.1789} {D=1 E=01 R=001 G=0001 A=00001 C+=00000 0.4642}
   {C="" 0.3571} {F=1 D=01 E=001 R=0001 G=00001 A=000001 C+=000000 0.6429}
   {C=1 F=01 D=001 E=0001 R=00001 G=000001 A=0000001 C+=0000000 1}

   Thus:
	Code	Length	Occurrences Bits Needed
   C    1	1	10	    10
   F    01	2	5	    10
   D    001	3	4	    12
   E    0001	4	3	    12
   R    00001	5	3	    15
   G    000001	6	1	    6
   A    0000001	7	1	    7
   C+   0000000	7	1	    7
   Total                28	    79

e. i.  79 bits are needed
   ii. Fixed length code required 84 bits, thus the huffman encoding is a little more efficient.
   iii. 2.5810 * 28 = 72.268; Not as efficient, obviously, not too far off either.