Connect an Independence Check to a Union
A free-response question may ask you to decide whether two events are independent and then find the probability that at least one occurs. The order matters: first check independence using the information provided; then use that result only where it is justified. A clear solution names the events, shows the relevant probabilities, and explains what each result means in context.
In Testing Independence with the Product Rule and Common Errors with Independence and Unions, you learned to compare a joint probability with the product of the marginal probabilities, and to account for the overlap when calculating a union. Here, we put those skills together in the style of a multi-part free-response question.
A probability-based independence check is not a significance test. It checks whether the probabilities in a stated chance model satisfy the definition of independence. If a question gives a two-way table of a defined group and says to select one member at random, the probabilities describe that selection process. A table from a sample, by itself, does not prove that the corresponding events are independent in a larger population.
A Reliable Plan for a Multi-Part Response
Write down what each event means before using a formula. Then identify what information is available: a two-way table, marginal and joint probabilities, or a conditional probability. The form of the information determines how you can check independence and calculate the overlap.
State what \(A\) and \(B\) represent and identify the chance process, such as selecting one record at random.
Compare \(P(A\cap B)\) with \(P(A)P(B)\), or compare \(P(A\mid B)\) with \(P(A)\) when \(P(B)>0\).
Use the joint probability from the information given. If independence is established, the product \(P(A)P(B)\) also gives the overlap.
Subtract the overlap from the sum of the two probabilities. State what “\(A\) or \(B\)” means in the context; it includes the possibility that both occur.
This sequence prevents a common error: using independence to calculate the overlap before checking whether the events are independent. If they are dependent, the general addition rule still works; use the actual joint probability or find it using an appropriate conditional probability.
Worked Example: Workshop Attendance and Compost Use
A community garden coordinator makes a hypothetical list of 240 volunteers. One record is selected at random. Let \(C\) be the event that the selected volunteer uses compost, and let \(W\) be the event that the volunteer attended a garden workshop. The table summarizes the list.
| Attended workshop | Did not attend | Total | |
|---|---|---|---|
| Uses compost | 30 | 42 | 72 |
| Does not use compost | 70 | 98 | 168 |
| Total | 100 | 140 | 240 |
The question asks whether \(C\) and \(W\) are independent and, whether or not they are, what is the probability the selected volunteer uses compost or attended a workshop?
State. The events are \(C=\) “the selected volunteer uses compost” and \(W=\) “the selected volunteer attended a workshop.” The chance process is selecting one of the 240 records at random.
Plan. Find the two marginal probabilities and the joint probability from the table. Compare the joint probability with the product of the marginal probabilities to check independence. Then find the union by adding the marginal counts and subtracting the overlap count.
Do. From the row and column totals,
The probability that both events occur is the count in the overlap cell divided by the grand total:
The product of the marginal probabilities is
The joint probability equals the product, so \(C\) and \(W\) are independent for this random selection from the listed volunteers. As a second check, \(P(C\mid W)=30/100=0.30=P(C)\).
For the union, the overlap must be counted only once. Using the table counts,
This agrees with the independent-events calculation:
Conclude. The two events are independent for this selection process because \(P(C\cap W)=P(C)P(W)\). The probability that a randomly selected volunteer uses compost or attended a workshop, including volunteers who did both, is about \(0.5917\).
When the Events Are Dependent
Independence is not required to calculate a union. The general addition rule applies to any pair of events. What changes is how you find the overlap: when events are dependent, do not substitute \(P(A)P(B)\) for \(P(A\cap B)\). Use the joint probability from the table, or use the general multiplication rule with a conditional probability, as in The General Multiplication Rule.
Worked Example: Delivery Method and On-Time Arrival
In a hypothetical set of 160 deliveries, 60 used an electric vehicle, 80 arrived within the promised time window, and 36 both used an electric vehicle and arrived on time. Select one delivery record at random. Let \(E\) be the event that it used an electric vehicle and \(T\) the event that it arrived on time. Check whether \(E\) and \(T\) are independent, then find the probability that a delivery used an electric vehicle or arrived on time.
State. \(E\) means “the selected delivery used an electric vehicle,” and \(T\) means “the selected delivery arrived on time.”
Plan. Calculate \(P(E)\), \(P(T)\), and \(P(E\cap T)\) using the total of 160 deliveries as the denominator. Compare the joint probability with the product. For the union, use the actual joint probability from the given overlap because independence may not hold.
Do. The marginal probabilities and joint probability are
If the events were independent, their joint probability would be
Since \(0.225\ne 0.1875\), \(E\) and \(T\) are not independent in this selection process. To check the same conclusion conditionally, \(P(E\mid T)=36/80=0.45\), which differs from \(P(E)=0.375\).
Use the observed overlap for the union:
A count-based check gives the same value: \(60+80-36=104\) deliveries satisfy at least one of the two conditions, and \(104/160=0.650\). The remaining 56 deliveries satisfy neither condition, so \(1-56/160=0.650\) is another check.
Conclude. The events are dependent because their joint probability differs from the product of their marginal probabilities. The probability that a randomly selected delivery used an electric vehicle or arrived on time, or did both, is \(0.650\).
Use Conditional Information to Find the Overlap
Sometimes a question provides a conditional probability instead of a joint probability. If \(P(B)>0\), the general multiplication rule gives \(P(A\cap B)=P(B)P(A\mid B)\). This gives the overlap needed for a union whether or not the events are independent. If the conditional probability equals the relevant marginal probability, that equality also provides an independence check.
Worked Example: Weather Alert and Equipment Check
For a hypothetical set of daily operations, let \(A\) be the event that a weather alert is issued and \(B\) the event that an equipment check is flagged for review. The probability of an alert is \(0.42\), the probability of a flagged check is \(0.35\), and \(P(A\mid B)=0.42\). Determine whether the events are independent and find \(P(A\cup B)\).
State. \(A\) means “a weather alert is issued,” and \(B\) means “an equipment check is flagged for review.”
Plan. Compare the given conditional probability \(P(A\mid B)\) with \(P(A)\) to check independence; \(P(B)=0.35>0\), so the comparison is valid. Then use the general multiplication rule to find the overlap and the general addition rule to find the union.
Do. The independence comparison is \(P(A\mid B)=0.42=P(A)\), so \(A\) and \(B\) are independent. Find the joint probability from the conditional information:
Now subtract that overlap from the sum of the event probabilities:
The product rule for independent events confirms the overlap: \(P(A)P(B)=(0.42)(0.35)=0.147\). Thus, the union calculation is consistent with the independence check.
Conclude. The events are independent because the probability of an alert among flagged checks equals the overall probability of an alert. The probability that a weather alert is issued or an equipment check is flagged, including days when both happen, is \(0.623\).
Common Mistakes and AP Exam Tips
- Checking the union instead of independence. Independence is checked by comparing a joint probability with the product of the marginal probabilities, not by comparing \(P(A\cup B)\) with a product.
- Multiplying without justification. \(P(A)P(B)\) is the overlap only when independence has been established or is part of the stated model. For dependent events, use the actual joint probability or a conditional probability.
- Forgetting the overlap in an inclusive “or.” The union includes outcomes in both events. Add the individual probabilities, then subtract \(P(A\cap B)\) once.
- Using the wrong denominator in a table. Marginal probabilities use the grand total. A conditional probability uses the total in the group named after the bar, as explained in Conditional Probability from a Two-Way Table.
- Making a claim broader than the chance process supports. If a table describes a particular list and one record is selected at random, describe independence for that selection process. Do not claim the table establishes independence for an entire population unless the problem supports that conclusion.
A full-credit response identifies both events, shows the numbers used for the independence comparison, states the conclusion, and gives the union calculation with the overlap subtracted. Finish with a sentence in context. For example: “Because the joint probability differs from the product of the marginal probabilities, the events are dependent for this selection process. Using the observed overlap, the probability of either event or both is \(0.650\).”
Key Takeaway
A multi-part probability response is easier to follow when each calculation has a clear purpose. Check independence with a joint-and-product comparison or a conditional-and-marginal comparison. Then use the overlap supported by the information to find the inclusive union.
Check Your Understanding
For each situation, show the key probability comparison or calculation and state what the result means.
- In a group of 100 randomly selectable records, 40 meet condition \(A\), 50 meet condition \(B\), and 20 meet both. Are \(A\) and \(B\) independent? Find \(P(A\cup B)\).
- Suppose \(P(R)=0.30\), \(P(S)=0.60\), and \(P(R\cap S)=0.15\). Check independence and calculate \(P(R\cup S)\).
- If \(P(M)=0.25\), \(P(N)=0.40\), and \(P(M\mid N)=0.25\), are \(M\) and \(N\) independent? Find \(P(M\cap N)\) and \(P(M\cup N)\).
- A two-way table gives a joint probability of \(0.18\), with marginal probabilities \(0.30\) and \(0.50\). What does the product-rule comparison say about independence? What additional calculation gives the union?
- Explain why the independent-events product cannot be used to calculate the overlap merely because a question asks for “\(A\) or \(B\).”