A research team has developed a face recognition device to match photos in a database. From laboratory tests, the recognition accuracy is 92% and trials are assumed to be independent.

(a) If the research team continues to run laboratory tests, what is the mean number of trials until failure?
(b) What is the probability that the first failure occurs on the tenth trial? Round your answer to four decimal places.
(c) To improve the recognition algorithm, a chief engineer decides to collect 6 failures. How many trials are expected to be needed? Round your answer to one decimal place

For each test, there are only two possible outcomes, either they are accurate, or they are not. Tests are independent, hence the binomial distribution is used to solve this question.

What is the binomial distribution formula?

The formula is:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

x is the number of successes.

n is the number of trials.

p is the probability of a success on a single trial.

The expected number of trials until q failures is:

[tex]E_f(X) = \frac{q}{1 - p}[/tex]

In this problem, the recognition accuracy is 92%, hence p = 0.92.

Item a:

One failure, hence q = 1 and:

[tex]E_f(X) = \frac{1}{0.08} = 12.5[/tex]

The mean number of trials until failure is of 12.5.

Item b:

First 9 successful, each with 0.92 probability, then the 10th is a failure, with 0.08 probability, hence:

[tex]p = 0.92^9 \times 0.08 = 00378[/tex]

There is a 0.0378 = 3.78% probability that the first failure occurs on the tenth trial.

Item c:

6 failures, hence q = 6 and:

[tex]E_f(X) = \frac{6}{0.08} = 75[/tex]

75 trials are expected to be needed.

More can be learned about the binomial distribution at https://brainly.com/question/14424710