An artificial intelligence company is developing software for determining whether a photograph contains a human face. Among all photographs that are processed by the software, 10% are known to contain a human face. For a photograph containing a human face, the probability that the software correctly determines the photograph contains a human face is 0.75 and for a photograph that does not contain any human faces, the probability that the software incorrectly determines the photograph contains a human face is 0.33. Suppose a randomly selected photograph is determined by the software to not contain any human faces. Find the probability that the photograph actually curtained a human face

Let's denote:

• F as the event that a photograph contains a human face
• NF as the event that a photograph does not contain a human face
• D as the event that the software determines a photograph contains a human face
• ND as the event that the software determines a photograph does not contain a human face

We are given:

• P(F) = 0.10 (probability that a photograph contains a human face)
• P(NF) = 1 - P(F) = 0.90 (probability that a photograph does not contain a human face)
• P(D|F) = 0.75 (probability that the software correctly determines the photograph contains a human face)
• P(D|NF) = 0.33 (probability that the software incorrectly determines the photograph contains a human face)

We are asked to find P(F|ND), the probability that a photograph contains a human face given that the software determines it does not contain a human face.

First, we need to find P(ND|F) and P(ND|NF), the probabilities that the software determines a photograph does not contain a human face given that it does or does not contain a human face, respectively.

Since the software either determines a photograph contains a human face or it does not, we have:

• P(ND|F) = 1 - P(D|F) = 1 - 0.75 = 0.25
• P(ND|NF) = 1 - P(D|NF) = 1 - 0.33 = 0.67

Now we can use Bayes' theorem to find P(F|ND):

P(F|ND) = P(ND|F) * P(F) / [P(ND|F) * P(F) + P(ND|NF) * P(NF)]

Substituting the given values:

P(F|ND) = 0.25 * 0.10 / [0.25 * 0.10 + 0.67 * 0.90] = 0.025 / [0.025 + 0.603] = 0.025 / 0.628 ≈ 0.0398

So, the probability that a photograph contains a human face given that the software determines it does not contain a human face is approximately 0.0398 or 3.98%.