A student considering two housing applications may reason that each has a 50% chance of success, so the chance of at least one offer must be 75%. That conclusion requires an additional assumption about how the outcomes relate. The two individual percentages alone do not determine their overlap.

OpenStax explains that multiplying event probabilities directly requires independence; in general, a conditional probability is needed. This article uses an invented count table to make that distinction visible. The numbers are not estimates for a real student, housing provider or college, and they do not establish that actual applications have known probabilities.

Keep the individual totals fixed

Imagine 100 equally weighted fictional cases. In every scenario below, application A succeeds in 50 cases and application B succeeds in 50 cases. What changes is how many cases receive both offers.

Outcome Independent-pattern example Same-outcome example No-overlap example
Both A and B 25 50 0
A only 25 0 50
B only 25 0 50
Neither 25 50 0
At least one 75 50 100

Each column sums to 100. In all three, A succeeds in 50 cases and B succeeds in 50. Yet the number with at least one offer is 75, 50 or 100. The separate success counts do not reveal the joint pattern.

Locate the extra assumption

In the independent-pattern example, the overlap is 50% × 50% = 25%. The probability of at least one is 50% + 50% − 25% = 75%. This is an illustrative model with independence stipulated, not a conclusion that real applications are independent.

In the same-outcome example, the same 50 cases receive both offers and the remaining 50 receive neither. In the no-overlap example, each successful case receives exactly one. The two edge cases show the arithmetic range compatible with these particular individual probabilities: at least one can range from 50% to 100%. They are not descriptions of plausible housing markets or estimates of actual dependence.

For general events A and B, count the cases in either group once: P(A or B) = P(A) + P(B) − P(A and B). If the intersection is not known, the sum alone double-counts cases with both. If the events are mutually exclusive, the intersection is zero; mutually exclusive is not the same as independent.

Ask what might connect the outcomes

Real applications may share application dates, documentation, budget constraints or a common eligibility criterion. An applicant's characteristics may affect both providers' decisions. Shared factors can create dependence, but identifying a possible connection does not tell you the direction or size of that dependence. A conditional rate such as P(B succeeds given A succeeds) requires relevant evidence for the same population and process.

A published overall offer rate is not automatically an individual student's success probability. Applicants differ and historical rules can change. Even a carefully sourced rate may describe a different applicant pool or a different stage of the process. Label its population, date and limitations before using it in any scenario.

Use scenarios as planning aids

The 50/75/100 comparison can help identify why a single unqualified number is fragile. It does not justify counting on any one result. For practical planning, record each provider's deadline, deposit requirement, cancellation terms and an alternative if neither offer arrives. Also consider whether accepting one offer prevents accepting the other; that is a decision rule after offers, not necessarily a statement about whether both could be offered.

A useful probability note shows the individual assumptions, the assumed overlap and the resulting calculation. If the overlap is unknown, say so. That leaves the student with a clearer view of what evidence is missing rather than a precise-looking chance unsupported by the two individual figures.