An IRS auditor randomly selects 3 tax returns from 59 returns of which 11 contain errors. What is the probability that she selects none of those containing errors? Round to four decimal places.
Events:
A=no error
B=with error
P(AAA)=(48/59)*(47/58)(46/57)
To find the probability that the IRS auditor selects none of the tax returns containing errors, we first need to find the total number of ways she can select 3 tax returns from the pool of 59 returns. Next, we calculate the number of ways she can select 3 tax returns that do not contain errors. Finally, we divide the number of successful outcomes by the total number of possible outcomes to get the probability.
Step 1: Calculate the total number of ways to choose 3 tax returns from 59 returns:
This is represented by the binomial coefficient, often denoted as "n choose k" or "C(n, k)".
In this case, n = 59 (total number of tax returns) and k = 3 (number of tax returns to be selected).
Using the formula for binomial coefficients, we have:
C(59, 3) = 59! / (3! * (59 - 3)!)
= 59! / (3! * 56!)
= (59 * 58 * 57) / (3 * 2 * 1)
= 165,725
So, there are 165,725 ways to choose 3 tax returns from the pool of 59 returns.
Step 2: Calculate the number of ways to choose 3 tax returns that do not contain errors:
Out of 59 returns, there are 11 tax returns that contain errors. Hence, there are 59 - 11 = 48 tax returns that do not contain errors. We need to calculate the number of ways to choose 3 tax returns from these 48.
Using the same formula as in Step 1, we have:
C(48, 3) = 48! / (3! * (48 - 3)!)
= 48! / (3! * 45!)
= (48 * 47 * 46) / (3 * 2 * 1)
= 17,296
So, there are 17,296 ways to choose 3 tax returns that do not contain errors.
Step 3: Calculate the probability:
The probability of the auditor selecting none of the tax returns containing errors is the ratio of the successful outcomes (17,296) to the total possible outcomes (165,725).
Probability = Successful outcomes / Total outcomes
= 17,296 / 165,725
≈ 0.1044 (rounded to four decimal places)
Therefore, the probability that the IRS auditor selects none of the tax returns containing errors is approximately 0.1044.