Final answer:
The probability that six employees who do not have asbestos in their lungs must be tested before finding the four who do have asbestos is approximately 9.99%.
Step-by-step explanation:
The question is asking for the probability that six employees without asbestos will be tested before finding four employees with positive indications of asbestos in their lungs. We can treat each employee test as an independent Bernoulli trial with a success probability (having asbestos) of 0.4 and a failure probability (not having asbestos) of 0.6. We are seeking the probability of having exactly 6 failures (no asbestos) followed by 4 successes (asbestos). We can use the negative binomial distribution to solve this problem.
The negative binomial distribution gives the probability of k successes and r failures in k+r trials, where the last trial must be a success. In this context, k equals 4 (four employees with asbestos), and r equals 6 (six employees without asbestos). The probability formula for the negative binomial distribution is:
P(X = r) = (r+k-1)C(k-1) * (success probability)^k * (failure probability)^r
Substituting the given values:
P(X = 6) = (6+4-1)C(4-1) * (0.4)^4 * (0.6)^6
Calculate the combination: (9)C(3) = 84 (ways to arrange 6 failures and 4 successes in 10 trials with the last trial being a success). Then, calculate the probability:
P(X = 6) = 84 * (0.4)^4 * (0.6)^6
Complete the calculation to find the probability:
P(X = 6) = 84 * 0.0256 * 0.046656
P(X = 6) ≈ 0.0999
Thus, the probability that six employees without asbestos will be tested before finding four employees with asbestos is approximately 0.0999 or 9.99%.