An IRS auditor randomly selects 3 tax returns from 55 returns of which six contain errors. What is the probability that she selects none of those containing errors?
To determine the probability that the IRS auditor selects none of the tax returns containing errors, we need to calculate the probability of selecting one return without an error, and then multiply it by itself for each return.
Let's break down the problem step by step:
Step 1: Calculate the probability of selecting a tax return without an error. Since there are 55 tax returns in total, and 6 contain errors, the probability of selecting a return without an error is:
P(Without Error) = (55 - 6) / 55 = 49 / 55
Step 2: Since the auditor selects 3 tax returns, we need to multiply the probability of selecting a return without an error by itself for each of the 3 returns:
P(Selecting None with Errors) = (49 / 55) * (49 / 55) * (49 / 55)
Step 3: Calculate the final probability:
P(Selecting None with Errors) = (49 / 55) * (49 / 55) * (49 / 55) ≈ 0.405
Therefore, the probability that the IRS auditor selects none of the tax returns containing errors is approximately 0.405, or 40.5%.
To calculate the probability of selecting none of the tax returns containing errors, we need to use the concept of combinations and the formula for the probability of an event.
Step 1: Determine the total number of ways to select 3 tax returns from 55 returns. This can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of tax returns and r is the number of tax returns to be selected. In this case, n = 55 and r = 3.
C(55, 3) = 55! / (3!(55-3)!) = 55! / (3!52!) = (55 x 54 x 53) / (3 x 2 x 1) = 22,770
Step 2: Determine the total number of ways to select 3 tax returns from the 6 returns containing errors. Similarly, we calculate:
C(6, 3) = 6! / (3!(6-3)!) = 6! / (3!3!) = (6 x 5 x 4) / (3 x 2 x 1) = 20
Step 3: Calculate the probability of not selecting any of the returns containing errors. This can be calculated by dividing the number of favorable outcomes (not selecting any returns containing errors) by the number of total outcomes.
Probability = Number of favorable outcomes / Number of total outcomes
Number of favorable outcomes = C(55-6, 3) = C(49, 3) = 49! / (3!(49-3)!) = (49 x 48 x 47) / (3 x 2 x 1) = 22,082
Therefore, the probability of not selecting any returns containing errors is:
Probability = Number of favorable outcomes / Number of total outcomes = 22,082 / 22,770 = 0.970
Initially, 49 out of 55 returns do not contain errors.
So probability of selecting the first (first trial) correct return is 49/55.
After this, there are 48 correct returns out of 54, so....48/54
For all three trials to succeed, we multiply the probabilities of all three trials to get the required probability.