Question

Gold has been discovered in the Worcester Hills! So far, 100 people have tried their hands at prospecting for gold in the Worcester Hills. 20% have found it and 80% have come up dry. The average value of finding gold (among those 20 people who found it) was $1,000,000 (that’s 1000 pazoozas, for those of you who are used to that form of currency.) You are thinking of trying your hand at prospecting, but it will cost you $250,000 in prospecting equipment, beer, and divorce proceedings.

1. Should you prospect and what is the EMV?

As fate would have it, Old Jack Nugget is a 95 year-old prospector who has scoured the Hills and “knows” where all the hidden gold is. He is willing to consult for you for $200,000 and direct you to the gold deposits. Of course, Jack is not perfect in his predictions and he also drinks a little: historically, he has been 80% accurate in predicting gold when there was gold and 90% accurate in predicting No Gold when there was No Gold.

2. Should you hire Jack?
3. What is the maximum consulting fee you would be willing to pay Jack?

Answer 1

Notation:

Gold = G

No Gold = NG

Predicting Gold = PG

Predicting No Gold = PNG

Economics:

Prospect Cost = $250,000

Consulting Cost = $200,000

Gold Profit = $1,000,000

Given Probabilities:

P(G) = 0.2

P(NG) = 0.8

P(PG|G) = 0.8

P(PNG|G) = 1 – P(PG|G) = 0.2

P(PNG|NG) = 0.9

P(PG|NG) = 1 - P(PNG|NG) = 0.1

Need:

P(PG)

P(PNG) = 1 – P(PG)

P(G|PG)

P(NG|PG) = 1 – P(G|PG)

P(G|PNG)

P(NG|PNG) = 1 – P(G|PNG)

P(PG) = P(PG|G)P(G) + P(PG|NG)P(NG)

P(PG) = (0.8)(0.2) + (0.1)(0.8) = 0.16 + 0.08 = 0.24

P(PNG) = 1 – P(PG) = 1 – 0.24 = 0.76

P(G|PG)P(PG) = P(PG|G)P(G)

P(G|PG) = P(PG|G)P(G)/P(PG)

P(G|PG) = (0.8)(0.2)/0.24 = 0.666667

P(NG|PG) = 1 – P(G|PG)

P(NG|PG) = 1 – 0.666667 = 0.333333333

P(G|PNG)P(PNG) = P(PNG|G)P(G)

P(G|PNG) = P(PNG|G)P(G)/P(PNG)

P(G|PNG) = (0.2)(0.2)/0.76 = 0.05263

P(NG|PNG) = 1 – P(G|PNG)

P(NG|PNG) = 1 – 0.05263 = 0.94737

Yes we hire Jack. The maximum consulting fee we should willing to pay will be $0.94737.