Bayes theorem in finance of a magical forest

by Kurious Fox

Here, we denote by \sim A the event NOT A.

Example 1: Magical Investment Returns

In the magical forest, gnomes invest in enchanted acorns, which sometimes turn into golden trees.

  • Probability of an acorn turning into a golden tree P(G)= 0.05
  • Probability of a positive return on investment if an acorn turns into a golden tree P(R|G)=0.95
  • Probability of a positive return on investment if an acorn does not turn into a golden tree P(R|\sim G)= 0.2

A gnome named Glim invests in an acorn and receives a positive return. What is the probability that the acorn has turned into a golden tree?

Answer:

Using Bayes’ Theorem:

P(G|R) = \frac{P(R|G) \cdot P(G)}{P(R|G) \cdot P(G) + P(R|\sim G) \cdot P(\sim G)}

Now,

  • P(R|G) = 0.95
  • P(G) = 0.05
  • P(R|\sim G) = 0.2
  • P(\sim G) = 1 - P(G) = 0.95

Therefore,

P(G|R) = \frac{0.95 \cdot 0.05}{0.95 \cdot 0.05 + 0.2 \cdot 0.95} = \frac{0.0475}{0.0475 + 0.19} = \frac{0.0475}{0.2375} \approx 0.2

So, Glim has about a 20% chance that the acorn has turned into a golden tree given the positive return.

Example 2: Dragons invest in enchanted gemstones

In the mystical forest of Eldoria, where elves sing and dragons soar, there’s a whimsical financial phenomenon known as “Dragonomics.” Dragons invest in enchanted gemstones, which have a chance of transforming into rare dragon hoards.

  • Probability of a gemstone transforming into a dragon hoard P(H)= 0.15
  • Probability of a positive return on investment if a gemstone transforms into a dragon hoard P(R|H)= 0.95
  • Probability of a positive return on investment if a gemstone does not transform into a dragon hoard P(R|\sim H)= 0.2

Now, a dragon named Flameheart invests in a gemstone and receives a positive return. What’s the probability that the gemstone has transformed into a dragon hoard?

Answer:

Using Bayes’ Theorem:

P(H|R) = \frac{P(R|H) \cdot P(H)}{P(R|H) \cdot P(H) + P(R|\sim H) \cdot P(\sim H)}

Now, we know that

  • P(R|H) = 0.95
  • P(H) = 0.15
  • P(R|\sim H) = 0.2
  • P(\sim H) = 1 - P(H) = 0.85

Hence,

P(H|R) = \frac{0.95 \cdot 0.15}{0.95 \cdot 0.15 + 0.2 \cdot 0.85} = \frac{0.1425}{0.1425 + 0.17} = \frac{0.1425}{0.3125} \approx 0.457

So, Flameheart has about a 45.7% chance that the gemstone has transformed into a dragon hoard given the positive return.

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