Final answer:
To find the probability that none of the tax returns will be audited, we can use the Poisson distribution to approximate the binomial distribution. The probability is approximately 0.643.
Step-by-step explanation:
To determine the probability that none of the tax returns will be audited, we can use the Poisson distribution to approximate the binomial distribution. The Poisson distribution is used when the number of trials is large and the probability of success is small. In this case, the number of tax returns sampled is 40 and the probability of being audited is 1.1%, so we can apply the Poisson distribution.
The Poisson distribution formula is given by P(x; λ) = (e^-λ * λ^x) / x!, where λ is the average number of occurrences, and x is the actual number of occurrences. In this case, λ = np, where n is the number of trials and p is the probability of success.
Using the given information, λ = 40 * 0.011 = 0.44. To find the probability that none of the tax returns will be audited, we substitute x = 0 into the Poisson distribution formula. P(0; 0.44) = e^-0.44 * 0.44^0 / 0! = e^-0.44 ≈ 0.643.
Therefore, the probability that none of the tax returns will be audited is approximately 0.643.