Recall from Problem 3.15 that \(30 \%\) of the Drosophila stocks at the BDSC are infected with Wolbachia. Suppose a research assistant randomly samples a stock one at a time until identifying an infected stock.

(a) Calculate the probability that an infected stock is found within the first 5 stocks sampled.

(b) What is the probability that no more than 5 stocks must be tested before an infected one is found?

(c) Calculate the probability that at least 3 stocks must be tested for an infected one to be found.

Data From Problem 3.15

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3.15 Wolbachia infection. Approximately 12,500 stocks of Drosophila melanogaster flies are kept at The Bloomington Drosophila Stock Center for research purposes. A 2006 study examined how many stocks were infected with Wolbachia, an intracellular microbe that can manipulate host reproduction for its own benefit. About 30% of stocks were identified as infected. Researchers working with infected stocks should be cautious of the potential confounding effects that Wolbachia infection may have on experiments. Consider a random sample of 250 stocks. (a) Calculate the probability that exactly 60 stocks are infected. (b) Calculate the probability that at most 60 stocks are infected. (c) Calculate the probability that at least 80 stocks are infected. (d) If a researcher wants to make sure that no more than 40% of the stocks used for an experiment are infected, does it seem reasonable to take a random sample of 250?