the consultant tells the investor interest rates will stay the same, what probability should the investor attach to the event that they actually will increase?

Question

An investor is unsure whether interest rates will increase (I), remain the same (S), or decrease (D) by mid-summer. Based on his own experience, he attaches the following probabilities to these events: P(I) = .4, P(S) = .3, and P(D) = .3. However, he decides to purchase the advice of a consultant. The consultant’s “track record” is summarized below.

The consultant’s “track record” is summarized below.

P(CI | I)=.7 P(CI | S)=.4 P(CI | D)=.2

remain the same (CS) P(CS | I)=.2 P(CS | S)=.5 P(CS | D)=.2

decrease (CD) P(CD | I)=.1 P(CD | S)=.1 P(CD | D)=.6

Answer 1

Zachros:

 

In the language of the problem, you want P(I | CS) = P ( I, CS) / P(CS) = P( CS | I)P(I)]

 

Now

 

P(I | CS) = P ( I, CS) / P(CS)   Formula for conditional probability,

 

P(CS) = P( CS | I)P(I) + P( CS | S)P(S) + P( CS | D)P(D)   Formula for total probability.

 

Also

 

= P ( I, CS)) = P( CS | I)P(I), again, using the formula for conditional probability.

 

The question has provided us the (marginal) prababilities P(I), P(S) and P(D) based on investor's experience.

 

Therefore,  we compute the answer as 

 

P(I | CS) = P( CS | I)P(I) / [P( CS | I)P(I) + P( CS | S)P(S) + P( CS | D)P(D) P(CS)]

 

             = [0.2 x 0.4] / {0.08 + 0.15 + 0.06} = 0.08 / 0.29 = 0.276.

 

Please check my computations and confirm the answer by posting a comment,  Thanks.

Dr. G.