In 2008, 1.01% of returns were audited, 85.6% returns received a refund, 49,907 the number examined resulting in a refund, 1,342,674 number examined resulting in no refund.
If a taxpayer receives a refund, what is the likelihood that the taxpayers return will be examined? How does this likelihood compare when the taxpayer is not to receive a refund?
I'm having a terrible time understanding probability and the setting up of this equation, any help or explanations would be greatly appreciated! Thank you
Please post the exact original text of the question. We need to examine the wording, even the punctuation to interpret correctly the given information.
Confirmation is needed for:
- is 85.6% out of the unaudited general population?
- Are 49097 and 1342674 the revised statistics as a result of the audit (because this proportion does not correspond to 85.6% of the unaudited returns) ?
In 2008, the IRS processed over 130 million tax returns. Most of these returns were individual tax
returns. The large majority of these individual returns received a refund due to overpayment of taxes. In
2008, 85.6% of individual returns
received a refund
The above wording is what I received. The 85.6% is where I'm stuck.
I know that the total returns examined is 1,391,581 as a result the refunds examined with a return is approx .0356
examined without a return is .9641
so would the returns not examined be .856 minus .0359 ????
Could you please post the whole question in its original form before you make your comments, please?
What is not clear is if the audit is selected/done only on those who got a refund, or a selected proportion (1.01%) of all the returns are selected at random for audit.
Assuming 1.01% of returns are selected at random from all returns,
# of returns audited = 1392581
percentage audited = 1.01%
Total # of returns = 1392581 / .0101 = 137879307
Number of unaudited returns
= 137879307-1392581
= 136486726
# of unaudited returns with refund = 136486726*0.856 = 116832637
# of unaudited returns without refund
= 136486726 - 136486726
= 19654089
--------- refunded----no refund -- total
audited---- 49907 --- 1342674 -- 1392581
unaudited - 116832637 - 19654089 - 136486726
Define events
R=refunded
A=audited
...
To find the likelihood that a taxpayer's return will be examined given they receive a refund, you need to compare the number of returns examined resulting in a refund to the total number of returns that received a refund.
First, let's calculate the number of returns examined resulting in a refund as a fraction of the total number of returns that received a refund.
The number of returns examined resulting in a refund is given as 49,907, and the total number of returns that received a refund is given as 85.6% of the total returns. We can calculate this value by multiplying the total returns by the percentage:
Total returns = (85.6/100) * total number of returns
Now we can calculate the likelihood by dividing the number of returns examined resulting in a refund by the total returns:
Likelihood = (number of returns examined resulting in a refund) / (total returns)
Likelihood = 49,907 / [(85.6/100) * total number of returns]
To find the likelihood when the taxpayer is not to receive a refund, we need to compare the number of returns examined resulting in no refund to the total number of returns that did not receive a refund.
The number of returns examined resulting in no refund is given as 1,342,674, and the total number of returns that did not receive a refund is calculated by subtracting the percentage of returns that received a refund from 100%.
Now we can calculate the likelihood in a similar way:
Likelihood = (number of returns examined resulting in no refund) / (total returns)
Likelihood = 1,342,674 / [(100 - 85.6)/100 * total number of returns]
By comparing the two calculated likelihoods, you can see how the likelihood of a return being examined changes depending on whether the taxpayer receives a refund or not.