Problem 26 - Algebra for College Students

Question

If the probability of a tax return not being audited by the IRS is 0.97, then what is the probability of a tax return being audited?

Step-by-Step Solution

Answer

The probability of a tax return being audited is 0.03.

Step 1 - Understand the Total Probability

The total probability of all possible outcomes must sum up to 1. In this case, there are two possibilities: a tax return can either be audited or not audited.

Step 2 - Define the Given Probability

The problem states that the probability of a tax return not being audited is 0.97.

Step 3 - Calculate the Probability of Being Audited

To find the probability of a tax return being audited, subtract the probability of not being audited from 1. This can be expressed mathematically as follows:\[ P(\text{audited}) = 1 - P(\text{not audited}) \]Substitute the given probability:\[ P(\text{audited}) = 1 - 0.97 \]Then, simplify the equation:\[ P(\text{audited}) = 0.03 \]

Key Concepts

total probability

When dealing with probabilities, it's important to remember that the probabilities of all possible outcomes must add up to 1. This is the principle of total probability. For example, in a situation where there are only two possible outcomes, such as getting audited or not getting audited, the total probability is the sum of the probability of both events: audited plus not audited.
In our exercise, since there are only two outcomes (audited or not audited), their probabilities must add up to 1. If you know the probability of one outcome, you can easily find the other by using this total probability concept.

complementary events

Complementary events are pairs of events where one event occurs if and only if the other does not. These events are mutually exclusive, meaning they cannot both happen at the same time. This concept is very useful in probability, especially when it's easier to calculate the probability of one event to find the probability of its complement.
In our example, the events 'getting audited' and 'not getting audited' are complementary. Given the probability of 'not getting audited' is 0.97, the probability of 'getting audited' can be found as its complement, because these two probabilities must add up to 1. Therefore, if you subtract the probability of 'not getting audited' from 1, you get the probability of 'getting audited'.

subtraction rule

The subtraction rule is a straightforward method used in probability to find the probability of one event when you know the probability of its complement. This rule simply states that you can subtract the probability of the complement event from 1 to find the probability of the event you’re interested in.
Mathematically, if we denote an event as A and its complement as A', then the subtraction rule can be written as: \[P(A) = 1 - P(A')\]
Using our previous example, if the probability of not being audited (A') is 0.97, then the probability of being audited (A) is: \[P(\text{audited}) = 1 - P(\text{not audited})\] \[P(\text{audited}) = 1 - 0.97 = 0.03\]
This straightforward approach helps in quickly finding the required probability whenever you're dealing with complementary events.