Joint Probability Bounds: What P(A) and P(B) Can Tell You

Marginals do not usually determine overlap

Let a=P(A), b=P(B) and j=P(A∩B). Without a specified relationship, the feasible overlap is max(0,a+b−1)≤j≤min(a,b). These are inclusive bounds, not a claim that every unknown overlap should be set to the midpoint. Additional information such as a known conditional probability or justified independence can identify one value. The marginals alone generally cannot. All probabilities must refer to the same sample space and lie between zero and one.

Derive the upper bound from containment

The overlap is a subset of A and also of B, so it cannot be larger than either event. Hence j≤min(a,b). For a=0.7 and b=0.6, the upper limit is 0.6. At that endpoint B can be entirely contained within A. The resulting only-A probability is 0.1, only-B is zero and neither is 0.3. This partition demonstrates that the endpoint is feasible rather than merely an inequality with no realizable event arrangement.

Derive the lower bound from total probability

The union is a+b−j and cannot exceed one. Rearranging gives j≥a+b−1. Overlap also cannot be negative, producing the lower limit max(0,a+b−1). With marginals 0.7 and 0.6, this is 0.3. Some overlap is forced because the two individual probabilities sum to 1.3. At j=0.3, only-A is 0.4, only-B 0.3 and neither zero; all disjoint regions are nonnegative and sum to one.

Translate overlap bounds into union bounds

Since union equals a+b−j, a larger overlap gives a smaller union when marginals stay fixed. The union range is max(a,b)≤P(A∪B)≤min(1,a+b). For 0.7 and 0.6, that is 0.7 through 1.0. Neither consequently ranges from zero through 0.3. Keep the direction reversal visible when substituting interval endpoints. Pairing the lower overlap with the lower union would mix incompatible scenarios rather than describe one consistent event arrangement.

Locate independence within the interval

If independence is explicitly assumed, j=ab=0.42 for these marginals. It lies within 0.3 through 0.6, and union is 0.88. This is one feasible relationship, not a consequence of feasibility alone. Choosing exclusivity would require j=0, which lies outside the interval and is impossible here. The bounds can therefore reject an inconsistent assumption even when they cannot select the correct remaining relationship from the available information.

Independence selects one interior scenario; it is not implied by the bounds.

Understand endpoint and zero-probability cases

If a=0, overlap is necessarily zero; if a=1, overlap equals b. Thus some special marginals do identify the joint probability. That does not make a conditional probability with zero conditioning denominator defined. For example, P(B|A) is undefined at a=0 even though the overlap is known to be zero. Keep joint-event existence and conditional-ratio validity separate. Boundary cases are useful tests for software because an indiscriminate divide or default zero can conceal these differences.

Check a proposed joint value

For marginals 0.7 and 0.6, a proposed j=0.2 makes neither equal 1−0.7−0.6+0.2=−0.1 and is invalid. A proposed j=0.65 exceeds the smaller event and makes only-B negative. These failures can arise from differing study populations, incompatible time periods, transcription errors or a mistaken conditional value entered as a joint one. Investigate the input meaning instead of clipping an impossible value to a boundary and presenting it as the supplied probability.

Use relationship-unknown mode

Enter 0.7 and 0.6 in the Probability Calculator and choose relationship unknown to obtain overlap bounds 0.3 through 0.6. The tool intentionally does not display one definitive overlap or union. Known-overlap mode can verify each feasible endpoint and the independence scenario separately. If a conditional probability is supplied, the website expects B given A and computes j=a×P(B|A). A conditional ratio and a joint probability are different inputs even when their numerical values happen to coincide.

Report an interval as the actual finding

Write the marginals, feasible overlap interval and which relationship information remains missing. These bounds are sharp for two arbitrary events: their endpoints can be attained by valid partitions such as those shown here. They do not supply a confidence interval for an estimated probability or account for sampling error in the marginals. If a and b themselves are uncertain estimates, that is an additional modeling problem. Preserve the distinction between logical feasibility and statistical estimation when communicating the result.