The probability that the total number of defects on the five bolts will be at least 6 is approximately 1.66%.
Let X be the number of defects on a bolt of cloth produced by the process. Since the number of defects follows a Poisson distribution with mean 0.4, the probability that there are X defects is given by:
P(X = x) = (0.4)^x * e^(-0.4) / x!
If we take a random sample of five bolts of cloth and let Y be the total number of defects on the five bolts, then the probability that there are at least 6 defects on the five bolts is given by:
P(Y >= 6) = 1 - P(Y <= 5)
We can use the cumulative distribution function (CDF) of the Poisson distribution to calculate the probability that there are at most 5 defects on the five bolts. The CDF of the Poisson distribution with mean λ is given by:
P(X <= k) = 1 - e^(-λ) * Σ_(x=0)^k (λ^x / x!)
Using this formula, we can calculate that:
P(Y <= 5) = 1 - e^(-0.4 * 5) * Σ_(x=0)^5 (0.4^x / x!) ≈ 0.9834
Therefore, the probability that there are at least 6 defects on the five bolts is:
P(Y >= 6) = 1 - 0.9834 ≈ 0.0166
So, the probability that the total number of defects on the five bolts will be at least 6 is approximately 1.66%.