Techniques

Question 1

Given P(FuR), P(Fc), P(F|R)=1-P(Fc)
Find P(R)

Answer 1

Given that P(F|R) = P(F)
This means P(FnR) = P(F)P(R)
P(FuR) = P(F) + P(R) - P(F)P(R)
P(FuR) - P(F) = (1-P(F))*P(R)
P(R) = (P(FuR) - P(F)) / P(Fc)

Question 2

Given P(N), P(M), P(NnM)
Find XOR of N and M

Answer 2

( P(N) + P(M) - P(NnM) ) - P(NnM)

Question 3

Given P(A), P(B|Ac)
Find P(AuB)

Answer 3

P(BnAc) = P(B|Ac)P(Ac)
P(AuB) = P(A) + P(BnAc)
P(AuB) = P(A) + P(B|Ac)P(Ac)

Question 4

Given P(Ac|Bc), P(AuB)
Find P(B)

Answer 4

P(AcnBc) = P(Ac|Bc)*P(Bc)
P(AcnBc) = 1- P(AuB)
P(B) = 1- [ [ 1- P(AuB) ] / P(Ac|Bc) ]

Question 5

Given P(F), P(A|F), P(A|Fc)

Answer 5

P(A) = P(AnF) + P(AnFc)
P(A) = P(A|F)P(F) + P(A|Fc)P(Fc)

Question 6

P(Q), P(G|Q), P(G|Qc)
Find P(Q|G)

Answer 6

P(G) = P(G|Q)P(Q) + P(G|Qc)P(Qc)
P(Q|G) = P(G|Q)*P(Q) / P(G)

Question 7

P(A|B) = P(Ac|Bc) = p, given P(B), P(B|A)
Find p

Answer 7

P(B|A) = P(AnB) / P(A)
P(AnB) = P(B)P(A|B) = P(B)p
P(BcnA) = P(A|Bc)P(Bc) = P(Bc)(1-p)
P(A) = P(Bc)(1-p) + P(B)p
P(A|B) = P(B)p / [ P(Bc)(1-p) + P(B)*p ]

Question 8

P(Ac|Bc)
Find P(A|Bc)

Answer 8

1-P(Ac|Bc)

Question 9

Given F(x)
Find E[rX+s]

Answer 9

f(x) = d/dx {F(x)}
E[x] = } x*f(x) dx
E[rX+s] = rE[X] + s

Question 10

Given F(x)
Find E[rX+s]

Answer 10

f(x) = d/dx {F(x)}
E[x] = } x*f(x) dx
E[rX+s] = rE[X] + s