What Does Independence Mean in a Real Setting?
In Is a Fixed Probability Realistic, we considered whether each trial could reasonably have the same probability of success. A separate question is whether the outcomes of different trials are independent. Independence is plausible when knowing the outcome of one trial would not change the chance of an outcome on another trial.
For example, imagine \(A\) is the event that one sibling catches a cold this week and \(B\) is the event that another sibling catches a cold this week. If the events are independent, knowing that \(A\) occurred does not change the probability of \(B\). In probability notation, independence means \(P(A\text{ and }B)=P(A)P(B)\). This relationship can help describe what an independence model predicts, but real settings require us to ask whether the model’s assumptions make sense.
Dependence does not require one outcome to directly cause another. Two outcomes can be connected because they share a cause, such as the same environment or weather system. Conversely, two events occurring close together in time are not automatically dependent; the relevant question is whether information about one outcome changes the probability of another.
Keep independence separate from a fixed success probability. A process can have the same probability of success on every trial but still have dependent outcomes. For instance, two people might each have the same chance of catching an illness, while a shared exposure makes their outcomes more likely to occur together. A process could also have independent outcomes but different probabilities from trial to trial. As discussed in Choosing Between Binomial and Other Models, a binomial model requires both independence and a common probability, along with the other binomial conditions.
Look for Connections Between Trials
In many real situations, independence is an assumption to evaluate, not a fact that can be established just by reading a short description. A practical assessment begins by asking what could connect one trial to another. Three common possibilities are carryover, shared influences, and the way observations are selected.
- Carryover or feedback: An earlier outcome changes a later trial. A machine may heat up after running, a participant may learn from an earlier attempt, or a person may change their behavior after an event.
- Shared influences: Several trials experience the same factor. Siblings may share a home or exposure; measurements on consecutive days may share weather conditions. A shared influence can make outcomes tend to occur together even if neither directly causes the other.
- Selection effects: The way trials are chosen can link outcomes. When people or objects are sampled without replacement from a limited group, choosing one changes what remains available for the next selection.
These are reasons to question independence, not automatic proof that trials are dependent. For example, two siblings may share a home but spend little time together during the period being studied. A weather pattern may persist across days, but the outcomes of interest might not respond much to it. Use details about the setting to explain why dependence is plausible, unlikely, or uncertain.
A Practical Independence Check
Use this sequence to organize an answer. It is especially useful when deciding whether a binomial model is reasonable. The independence judgment should be based on the described process, not just on the fact that several trials are being counted.
Specify what is repeated and what counts as success. Without a clear trial, it is difficult to identify what could connect trials.
Ask whether an earlier result could change a later result or the conditions under which the next trial occurs.
Consider common environments, time patterns, family or group membership, and whether trials are selected without replacement from a limited population.
State whether independence seems plausible, questionable, or impossible to assess from the information given. Name the specific feature supporting your judgment.
If the setting involves a random sample taken without replacement, the 10% condition from The 10% Condition for Independence in Binomial Settings is a useful check: the sample should be no more than 10% of the population for independence to be a reasonable approximation. Passing that check supports the approximation; it does not guarantee that every other part of a model is appropriate.
Worked Example: Outcomes for Siblings
Worked Example: Outcomes for Siblings
A fictional health educator considers a model for whether two siblings develop a mild respiratory symptom during the same week. The model assigns each sibling a probability of 0.30 for developing the symptom. An illustrative joint model assigns probability 0.15 to both siblings developing it. Assess whether treating the siblings’ outcomes as independent is reasonable under this model.
State. Let \(A\) be the event that the first sibling develops the symptom, and let \(B\) be the event that the second sibling develops it. The proposed marginal probabilities are \(P(A)=0.30\) and \(P(B)=0.30\).
Plan. If the outcomes were independent, the probability that both siblings develop the symptom would be the product of their individual probabilities. Compare that prediction with the joint probability specified in the example, then consider whether the context offers plausible shared influences.
Do. Under independence, the predicted probability of both events is:
The illustrative joint model gives \(P(A\text{ and }B)=0.15\), which is not the independence prediction of 0.09. The difference indicates that the specified joint model does not treat the events as independent. This is a hypothetical model, not a claim about actual siblings.
The setting also offers a plausible explanation to investigate: siblings may share a home, contact with one another, or exposure to the same people. Such shared circumstances could make their outcomes more likely to occur together. The example does not show which factor matters or establish a cause; it shows why independence should not be assumed automatically.
Conclude. Under the illustrative joint model, the siblings’ outcomes are not independent because the probability of both developing the symptom is 0.15 rather than the 0.09 predicted by independence. In a real setting, shared exposure is a reason to question independence, though more information would be needed to assess the model.
This example also illustrates why equal marginal probabilities do not establish independence. Both siblings have the same stated individual probability, but their outcomes can still be more likely to occur together. A common \(p\) and independent trials are distinct assumptions.
Worked Example: Rain on Consecutive Days
Worked Example: Rain on Consecutive Days
A fictional planning exercise assigns a 0.30 probability of rain to each of two consecutive days. Its weather model assigns a 0.18 probability that it rains on both days. Assess whether the two daily rain outcomes are independent and explain what the calendar sequence alone tells you.
State. Let \(A\) be the event that it rains on the first day and \(B\) the event that it rains on the second day. The exercise specifies \(P(A)=0.30\), \(P(B)=0.30\), and \(P(A\text{ and }B)=0.18\).
Plan. Compare the specified probability of rain on both days with the product that independence would predict. Then interpret the result in light of the fact that consecutive days may share a weather system.
Do. If the daily outcomes were independent, the probability of rain on both days would be:
The model specifies 0.18 for rain on both days, rather than 0.09. Therefore, the two rain events are not independent under the stated model. The model gives the second day’s rain outcome a connection to the first day’s outcome.
The outcomes being on consecutive days does not, by itself, prove dependence. Here, the stated joint probability establishes dependence within the hypothetical model. In an actual weather setting, a weather system continuing across days would be a plausible shared influence to consider.
Conclude. The two daily rain outcomes are dependent under this exercise’s model: the probability of rain on both days is 0.18, while independence would predict 0.09. The calendar sequence alone would not be enough to reach that conclusion.
When a setting supplies only individual probabilities and no joint information, you generally cannot calculate or verify independence from those probabilities alone. Instead, assess the process and explain which details make independence plausible or questionable.
Worked Example: Selecting Students Without Replacement
Worked Example: Selecting Students Without Replacement
A fictional school has 120 students, of whom 36 are members of a music club. A student organizer selects 30 students without replacement to receive a survey. Assess the independence assumption for the selection trials. To see how selecting one student can affect the next selection, compare the probabilities for the first two selections.
State. A success is selecting a music-club member. The organizer makes 30 selections from a group of 120, without returning each selected student to the group.
Plan. Because selection is without replacement, the number of club members remaining can depend on earlier selections. Check the 10% condition for treating the selections as approximately independent, and calculate the probability of a club member on the second selection in two different cases.
Do. The sample is 30 students from a population of 120. Ten percent of 120 is 12, and \(30>12\), so the sample exceeds 10% of the population. The 10% condition is not met; independence is not well supported as an approximation by that condition.
For the first selection, the probability of choosing a club member is \(36/120=0.30\). If the first student selected is a club member, then 35 club members remain among 119 students, so the probability that the second student is a club member is:
If the first student selected is not a club member, then all 36 club members remain among 119 students, so the probability that the second student is a club member is:
The second-selection probability depends on the first outcome: it is about 0.2941 after selecting a club member, and about 0.3025 after selecting a nonmember. Thus, the outcomes are not exactly independent. The 10% condition also warns against treating this sample as approximately independent for a binomial model.
Conclude. The selection outcomes are dependent because each student is removed before the next selection, changing the number of club members and total students remaining. Since the sample of 30 is more than 10% of the population of 120, the 10% condition does not support an independence approximation either.
When a sample is small relative to a much larger population, the change caused by removing one selected individual may be negligible for the question at hand. That is why the 10% condition is useful. It is a practical guideline for approximation, while selection without replacement still creates an exact connection between selections.
Common Mistakes and AP Exam Tips
- Assuming repeated trials are automatically independent. Repeating the same kind of measurement does not remove carryover or shared influences. Full-credit reasoning identifies a feature of the process that could connect the outcomes.
- Confusing independence with a common probability. Equal probabilities for each trial do not guarantee independence. Explain separately whether the chance stays the same and whether earlier outcomes affect later ones.
- Claiming that shared circumstances prove dependence. A common home, team, or time period may be relevant, but a careful answer says it is a plausible source of dependence unless the model or data establish a connection.
- Thinking dependence always means direct cause and effect. A shared influence can make outcomes associated without either outcome causing the other. Name the possible common influence rather than asserting a direct cause without evidence.
- Ignoring sampling without replacement. In a limited population, each selection changes what remains. Check the 10% condition when considering a binomial approximation, and state whether it is satisfied.
- Overstating what the information shows. If a scenario gives no joint probabilities or details about the process, do not claim to have proved independence or dependence. Say what can be assessed and what remains unknown.
Key Takeaway
Independence is a claim about whether one outcome changes the chance of another. Carryover, shared influences, and sampling without replacement are reasons to examine that claim. They do not all lead to the same conclusion in every setting, so use the scenario’s specific details and state a qualified judgment.
Check Your Understanding
For each scenario, identify a possible connection between outcomes and give a cautious judgment about independence.
- Two siblings attend the same crowded event. Explain one reason their chances of developing a symptom afterward might be connected, and why that possibility does not prove dependence.
- A student makes the same type of free throw ten times. Name one feature that could make later attempts dependent on earlier attempts.
- A weather model gives each of two days a rain probability of 0.40 and a probability of 0.16 that both days are rainy. Are the events independent under this model? Show the comparison that supports your answer.
- A club has 200 members among 2,000 students. A random sample of 100 students is selected without replacement. Does the sample meet the 10% condition? What does that suggest about using independence as an approximation?
- Can trials have the same probability of success but dependent outcomes? Explain using a plausible shared influence.