An East Coast manufacturer of printed circuit boards exposes all finished boards to an online automated verification test. During one period, 900 boards were completed and 890 passed the test. The test is not infallible. Of 30 boards intentionally made to have noticeable defects, 25 were detected by the test. Use the relative frequencies to approximate the conditional probabilities needed below.

(a) Give an approximate value for \(P[\) Pass test \(\mid\) board has defects].

(b) Explain why your answer in part a may be too small.

(c) Give an approximate value for the probability that a manufactured board will have defects. In order to answer the question, you need information about the conditional probability that a good board will fail the test. This is important to know but was not available at the time an answer was required. To proceed, you can assume that this probability is zero.

(d) Approximate the probability that a board has defects given that it passed the automated test.