Sheldon Ross Introductory Statistics - Probability Section 4.6

Bayes' Theorem

1.
F-> event that the fair coin is selected
B-> event that the biased coin is selected

(a) P(Heads) = P(H|F).P(F) + P(H|B).P(B)
= 0.5 * 0.5 + 0.6 * 0.5
= 0.55

(b) P(F|T) = P(F∩T)/P(T) = P(T|F)P(F)/P(T)

= P(T|F).P(F) / (P(T|F).P(F) + P(T|not F).P(not F))

= (0.5 * 0.5) / (0.5 * 0.5 + 0.4 * 0.5) = 0.555555556

2.
K-> event that the student knew the answer
C-> event that the student answered it correctly

P(K|C) = P(C|K). P(K) / (P(C|K). P(K) + P(C|not K). P(not K))

= (1 * 0.6) / (1 * 0.6 + 0.2 * 0.4 ) = 0.882352941

3.
L -> event that the suspect is left handed
G -> event that the suspect is guilty

(a) P(L) = P(L|G). P(G) + P(L|not G).P(not G)
= 1 * 0.6 + 0.18 * 0.4 = 0.672

(b) P(G|L) = P(L|G).P(G) / (P(L|G).P(G) + P(L|not G).P(not G))

= (1 * 0.6 ) / ( 1 * 0.6 + 0.18 * 0.4 ) = 0.892857143

4.
U1 - 4R 3B
U2 - 2R 2B
R1 -> event that the ball drawn from urn 1 was red
B1 -> event that the ball drawn from urn 1 was blue
R2 -> event that the ball drawn from urn 2 is red
B2 -> event that the ball drawn from urn 2 is blue

If R1 happened, there are 3R 2B in U2. Note that, R1 = not B1

(a) P(R2) = P(R2|R1).P(R1) + P(R2|not R1).P(not R1)
= (3/5)*(4/7) + (2/5)*(3/7)
= 0.514285714

(b) P(R1|B2) = (B2|R1) . P(R1) / ((B2|R1) . P(R1) + P(B2|not R1) . P(not R1) )
= (2/5)* (4/7) / ( (2/5)* (4/7) + (3/5) * (3/7) )
= 0.470588235

5.
P-> person's diagnosis is positive
D-> person has the disease

P(D|P) = P(P|D).P(D)/( P(P|D).P(D) + P(P|not D).P(not D) )

= (0.97 * 0.02) / (0.97 * 0.02 + 0.03 * 0.98)
= 0.397540984

6.
R -> this side of the card is red
M -> this is the mixed card (bi coloured)

P(M|R) = P(R|M).P(M) / (P(R|M).P(M) + P(R|not M).P(not M))

= (0.5 * 1/3) / ( (0.5 * 1/3) + (0.5 * 2/3) )
= 0.333333333

7.
(a) P(in favour) = P(A|D) + P(A|R) (A, D, R self explanatory)

= 0.42 * 0.48 + 0.64 * 0.52 = 0.5344

(b) P(R|not A) = P(not A|R) * P(R)/( P(not A|R) * P(R) + P(not A|not R) * P(not R) )

= (0.36 * 0.52) / (0.36 * 0.52 + 0.58 * 0.48)
= 0.402061856

9.
C-> it's a California household
T-> earns over 250k/year

(a) Let x be the total no of households in the US

No of households in Cal = 0.12x
No of households in Cal earning over 250k/year = 3.3 % of 0.12x
= 0.00396x

No of households in the US earning over 250k/year = 0.013x

Hence, no of households in non-Cal earning over 250k/year = 0.013x - 0.00396x
= 0.00904x

The same thing as a percentage = 0.00904x/total non-Cal households
= 0.00904x/0.88x
= 0.010272727
~ 0.0103

(b) P(C|T) = P(T|C). P(C) / (P(T|C). P(C) + P(T|not C). P(not C))

= 0.033 * 0.12 / (0.033 * 0.12 + 0.0103 * 0.88)

= 0.304054054
~ 0.3046