Independence Is an Assumption About a Chance Process
In Using Venn Diagrams to Test Independence, you compared \(P(A\cap B)\) with \(P(A)P(B)\). In real situations, though, the probabilities may not be fully known. Often the practical question is whether it is reasonable to model two events as independent. That is a judgment about how the chance process works, not simply a calculation from a diagram.
Recall from What Independent Events Mean that events are independent when knowing that one occurred does not change the probability of the other. In a real context, ask what information about the chance process is revealed when you learn that one event happened. Could it tell you something about a shared person, place, time, or cause? If so, independence may be questionable.
An independence assumption does not mean that two events have nothing in common. It means the probability of one is unchanged by knowing whether the other occurred. Events can concern the same broad subject and still be independent, or concern different subjects but be dependent because of a shared cause.
The first step is to define the events and the chance process precisely. “It rains” is incomplete if it is unclear whether the event is rain today, rain at a particular location, or rain at any time during a week. Likewise, “a family member gets sick” needs a specified person or group and time period. Independence depends on those definitions and on the population or process under consideration.
Look for Shared Causes and Connections
A useful real-world check is to ask whether the events share a factor that can affect both. The factor might be a common environment, exposure, schedule, device, or time period. Learning that one event occurred can provide information about that factor, which may in turn change the probability of the other event.
- Shared environment: People in the same household may encounter the same germs, food, or air quality.
- Nearby times: Weather on one day may provide information about the weather on the next day because conditions often persist.
- Shared mechanism: Two outputs from the same machine or production batch may be connected by the machine’s current condition.
- Separate mechanisms: Outcomes from separate, stable chance mechanisms may be reasonably modeled as independent, if the process gives no important connection between them.
These are clues, not automatic rules. Family members will not always have the same outcome, and consecutive days will not always have the same weather. The question is whether the probability of one event changes after we learn about the other. A common influence makes such a change plausible, so it deserves attention.
Sometimes independence is a useful approximation even when perfect independence is unlikely. For example, an influence may exist but be weak compared with the variation in outcomes. In that case, an independence model may simplify a calculation. A careful explanation should still identify the assumption and say why it seems acceptable for the question being asked.
Worked Examples
Worked Example: Illness Among Siblings
Suppose a probability model for a particular winter week assigns each of two siblings a \(0.12\) probability of developing a cold. Let \(A\) be the event that the older sibling develops a cold, and \(B\) the event that the younger sibling develops a cold. A family health planner considers the independence model and estimates that \(P(B\mid A)=0.30\). Is treating the events as independent reasonable according to these probabilities?
State. The events concern two different people in the same household during the same week. We will compare the conditional probability of the younger sibling developing a cold when the older sibling has one with the younger sibling’s unconditional probability.
Plan. If the events were independent, learning that \(A\) occurred would not change the probability of \(B\). Thus, independence would require \(P(B\mid A)=P(B)=0.12\). We can also use the general multiplication rule to find the joint probability from the supplied conditional probability, then compare it with the product required by independence.
Do. The model gives \(P(B\mid A)=0.30\), while \(P(B)=0.12\). These are not equal. The conditional probability is higher by \(0.30-0.12=0.18\), or 18 percentage points.
Using the general multiplication rule, the probability that both siblings develop a cold is
As an arithmetic check, \(12\%\) of \(30\%\) is \(0.12\times0.30=0.036\), or \(3.6\%\). If the events were independent, the multiplication rule for independent events would instead give
That product is \(1.44\%\), since \(12\times12=144\) and the two factors together have four decimal places. The model’s joint probability, \(3.6\%\), is not the independence-model value of \(1.44\%\).
Conclude. According to the supplied probabilities, the events are dependent, so independence is not a reasonable description of this model. The shared household could provide a plausible connection, such as exposure to the same germs. The conclusion concerns this specified week and these two siblings; it does not mean that the siblings must both get sick whenever one does.
Worked Example: Rain on Consecutive Days
For a hypothetical location and season, let \(A\) mean that it rains on Tuesday and \(B\) mean that it rains on Wednesday. Suppose the model gives \(P(A)=0.30\), \(P(B)=0.30\), and \(P(B\mid A)=0.60\). Should a forecaster treat the two rain events as independent for this model?
State. The events concern rain at the same location on consecutive days. We will determine whether knowing it rained Tuesday changes the modeled probability of rain Wednesday.
Plan. Independence would require \(P(B\mid A)=P(B)\). We will compare those probabilities and, as a second check, compare the joint probability from the general multiplication rule with the product of the two marginal probabilities.
Do. The probability of Wednesday rain given Tuesday rain is \(0.60\), while the overall probability of Wednesday rain is \(0.30\). Because \(0.60\ne0.30\), the modeled probability changes when Tuesday rain is known.
The probability of rain on both days is
The multiplication checks as \(30\times60=1800\), with four decimal places in the factors, giving \(0.1800=0.18\). Under an independence assumption, the joint probability would be
Here \(30\times30=900\), which gives \(0.0900=0.09\). The two-day joint probability in the model, \(0.18\), differs from the independence-model value, \(0.09\).
Conclude. The events are dependent in this model, so independence is not appropriate for these probabilities. Consecutive days can share weather conditions, making rain on Tuesday informative about Wednesday. This is a reason to question independence, not a claim that consecutive weather outcomes are always dependent by exactly the same amount in every place or season.
Worked Example: Two Digits from a Randomizer
A classroom app is designed to generate two digits, each from 0 through 9, using a new random draw for each digit. Assume each digit is equally likely and that the app’s two draws use separate random steps. Let \(A\) mean the first digit is 7, and \(B\) mean the second digit is even. Is independence a reasonable model for these events?
State. The two events concern different draws from the app. We will use the stated design assumptions to find the individual and joint probabilities, then consider whether the mechanism gives a reason for one draw to affect the other.
Plan. If the digits are equally likely, one of ten digits is 7, and five of ten digits are even. If each draw is separate and the first result does not alter the second draw’s chances, the independence product rule is reasonable. We will calculate the model’s joint probability and compare it with the product of the marginal probabilities.
Do. There is one favorable digit out of ten for \(A\), so \(P(A)=\frac{1}{10}=0.10\). There are five even digits, \(0,2,4,6,8\), out of ten, so \(P(B)=\frac{5}{10}=0.50\). Under the stated independence assumption,
As a check, \(\frac{1}{10}\times\frac{5}{10}=\frac{5}{100}=\frac{1}{20}=0.05\). The model assigns a \(5\%\) probability to the first digit being 7 and the second digit being even.
The separate-draw design gives a reason to regard the events as independent: the first digit is not removed from a fixed set of ten digits before the second draw, and the problem states that the second random step does not use the first result. If the app instead selected two digits without replacement from a set in which each digit appeared only once, the first draw could change what was available for the second, and the independence assumption would need to be reconsidered.
Conclude. Independence is a reasonable model under the stated design: equally likely digits are generated in separate random steps, and the first result does not affect the second step. The conclusion relies on those assumptions about the app, not merely on the fact that the two digits are written in different positions.
Common Mistakes and AP Exam Tips
- Assuming different people or events must be independent. Two siblings are different people, but they may share exposures. A full-credit explanation considers whether learning one event could change the probability of the other.
- Assuming events are dependent just because they happen near each other. Events on consecutive days deserve scrutiny, but timing alone does not establish dependence. Explain a plausible connection or use supplied conditional probabilities to show that the probability changes.
- Confusing a possible shared cause with proof. A household, location, or common device can create a connection, but the presence of a shared setting alone does not quantify how much probabilities differ. State the reason the assumption is questionable without claiming certainty beyond the information given.
- Treating a convenient model as a real-world fact. A calculation using \(P(A)P(B)\) depends on independence. Say that you are assuming independence and briefly justify it; do not present the result as guaranteed if the assumption is uncertain.
- Leaving the events vague. “Weather is independent” is not a clear claim. Name the events, location, and time period. Independence is a relationship between specified events in a specified chance process.
Key Takeaway
Independence is a model of how events are related. To judge whether it is reasonable, define the events and look for shared influences or changes in the chance process. Family members and nearby weather days may be connected by shared conditions; separate random steps may support an independence model when one result does not affect the other.
Check Your Understanding
For each situation, identify a reason independence may be reasonable or questionable, and state what additional information would help if needed.
- Two people in the same household are selected. Let \(A\) mean the first person has a cold this week and \(B\) mean the second person has a cold this week. What shared influence might make independence questionable?
- Let \(A\) mean it snows on one day and \(B\) mean it snows the next day in the same town. Explain why consecutive timing is relevant, without claiming that it alone proves dependence.
- A randomizer generates two digits in separate steps, with each digit equally likely to be 0 through 9. What design feature would support treating the digits as independent?
- Suppose \(P(B)=0.25\) and \(P(B\mid A)=0.40\). Are \(A\) and \(B\) independent? Explain what the comparison means.
- Why should an answer specify the people, location, or time period when discussing independence in context?