From Program Probabilities to a Portfolio Probability
A development portfolio may contain several programs, each with its own probability of success. A decision-maker may want a different quantity: the probability that at least one of those programs succeeds. That portfolio-level probability is not generally the sum of the individual probabilities. The same favorable outcome can contribute to more than one program, and programs may share sources of risk.
The calculation is useful for describing the chance of obtaining at least one successful asset from a set of programs. It does not, by itself, say whether the portfolio is economically attractive, whether a particular allocation is optimal, or how much value a successful program creates. Those questions require additional decision inputs.
The success criterion and time horizon must be consistent across the calculation. For example, “technical success at the end of a trial” and “regulatory approval within five years” are different events. The \(p_i\) values should refer to the events actually being combined. If a program-level probability already includes the chance of operational failure, do not apply that same operational-failure adjustment again.
The Independent-Programs Calculation
If program outcomes are mutually independent, the probability that all programs fail is the product of their failure probabilities. Since “at least one succeeds” is the complement of “all fail,” the portfolio probability is one minus that product.
Mutual independence is stronger than saying that every pair of programs appears unrelated. It means that the joint probabilities factor appropriately for every collection of program events. In practice, the assumption is a modeling choice that needs justification; it does not follow merely from the programs having different names, targets, or teams.
Worked Example: Three Independent Programs
Suppose three hypothetical programs have probabilities of meeting their respective success criteria of \(0.30\), \(0.20\), and \(0.10\) over the same stated horizon. Assume their success events are mutually independent. Find the probability that at least one succeeds.
The corresponding failure probabilities are \(1-0.30=0.70\), \(1-0.20=0.80\), and \(1-0.10=0.90\). Under independence, the probability that all three fail is \(0.70\times0.80\times0.90=0.504\).
Therefore, \(P(\text{at least one success})=1-0.504=0.496\). The hypothetical portfolio has a \(49.6\%\) probability of at least one success under the stated independence assumption. This is not the sum \(0.30+0.20+0.10=0.60\); that sum counts some possible outcomes more than once.
Why Dependence Changes the Answer
Without independence, the individual probabilities \(p_i\) do not determine the probability of at least one success. The exact complement formula still applies, but the joint probability that all programs fail must be known or modeled. For two programs, inclusion-exclusion gives an especially direct expression: add the two marginal probabilities and subtract the probability that both succeed. Subtraction is needed because the joint-success event is included in both marginal probabilities.
For three programs, the same inclusion-exclusion principle adds the three marginal probabilities, subtracts the three pairwise joint-success probabilities, and adds the probability of all three succeeding. For larger portfolios, this pattern continues through intersections of increasing size. If only the marginal probabilities are available, using the independent-program formula implicitly supplies missing joint information: it assumes those joint outcomes factor as they would under independence.
Worked Example: Two Dependent Programs
Consider two hypothetical programs with \(p_1=0.40\) and \(p_2=0.30\). Suppose their joint probability of success is \(P(S_1\cap S_2)=0.18\). Find the probability that at least one succeeds, and compare it with the independent calculation.
Using inclusion-exclusion, \(P(S_1\cup S_2)=0.40+0.30-0.18=0.52\). Thus, the probability of at least one success is \(52.0\%\). The joint probability is feasible because it is no greater than either marginal probability and no less than zero.
If independence were assumed instead, the probability of both succeeding would be \(0.40\times0.30=0.12\), giving \(0.40+0.30-0.12=0.58\). The supplied joint probability, \(0.18\), is larger than the independent value, so successes co-occur more often in this example. That leaves fewer outcomes in which exactly one program succeeds, and the at-least-one probability is lower: \(0.52\), not \(0.58\).
This comparison illustrates why “positive dependence” should not be interpreted as “a higher chance of at least one success.” Dependence that concentrates successes in the same scenarios can increase the chance of multiple successes while decreasing the chance of obtaining any success relative to an independent calculation with the same marginals.
Dependence Through a Shared Condition
A useful way to represent dependence is to identify a shared condition that affects multiple programs. Conditional on that condition, outcomes may be modeled as independent even though they are dependent overall. The unconditional portfolio probability is then an average of the conditional portfolio probabilities, weighted by the probabilities of the shared conditions. This approach is only as credible as the conditions and conditional probabilities specified in the model.
Worked Example: A Shared Favorable or Unfavorable Environment
Suppose three hypothetical programs face the same uncertain development environment. A favorable environment occurs with probability \(0.40\); an unfavorable environment occurs with probability \(0.60\). Conditional on the environment, assume the program outcomes are independent. Each program has a \(0.60\) success probability in the favorable environment and a \(0.10\) success probability in the unfavorable environment.
For any one program, the marginal success probability is \(0.40\times0.60+0.60\times0.10=0.30\). If one ignored the shared environment and treated the three marginal outcomes as independent, the calculation would be \(1-(1-0.30)^3=1-0.343=0.657\).
Instead, calculate the chance of at least one success within each environment. In the favorable environment it is \(1-(1-0.60)^3=1-0.064=0.936\). In the unfavorable environment it is \(1-(1-0.10)^3=1-0.729=0.271\). Averaging over the environments gives \(0.40\times0.936+0.60\times0.271=0.3744+0.1626=0.537\). The model therefore gives a \(53.7\%\) probability of at least one success, not \(65.7\%\). The programs share exposure to a condition that tends to make their outcomes succeed or fail together.
The same method works for more than two shared scenarios: calculate the at-least-one probability conditional on each scenario, then sum those probabilities weighted by the scenario probabilities. This is not a claim that the scenario model is uniquely correct. It is a transparent way to state how a particular representation of dependence changes the result.
Bounds, Checks, and Common Failure Modes
When exact joint probabilities are unavailable, marginal probabilities still give simple bounds. The probability of at least one success cannot be smaller than the largest individual success probability, because success of that program alone guarantees at least one success. It cannot exceed the sum of the marginal probabilities, or 1 if that sum is larger than 1.
The upper bound follows from the union bound: adding the probabilities counts each outcome with one or more successes at least once, and may count it multiple times. The lower bound follows because \(S_i\subseteq S_1\cup\cdots\cup S_m\) for every \(i\). These bounds can be wide; they are checks on an answer, not necessarily useful estimates of the exact probability.
Worked Example: Bounding a Portfolio Probability
For three hypothetical programs, suppose the marginal probabilities of success are \(0.25\), \(0.40\), and \(0.15\), but no joint outcome information is available. Find valid bounds for the probability of at least one success.
The lower bound is the largest marginal probability: \(\max(0.25,0.40,0.15)=0.40\). The sum is \(0.25+0.40+0.15=0.80\), which is below 1, so the upper bound is \(0.80\). Therefore, \(0.40\leq P(\text{at least one success})\leq0.80\). The information given does not justify choosing a single value within this interval, including the independence estimate.
A practical calculation should be checked for both event definition and probability logic. Are all programs being evaluated against compatible success criteria and time horizons? Are the program probabilities marginal probabilities for those events, or are they conditional on a scenario? Has a shared operational or technical risk already been included? Does the result lie between the largest marginal probability and the union bound? If an independence model is used, is there a reason to expect the joint failure probability to be the product of the marginal failure probabilities?
A common mistake is to add all program success probabilities and call the sum the chance of at least one success. For the first example, this gives \(60.0\%\), whereas the independence calculation gives \(49.6\%\). The sum is only an upper bound when it is at most 1; it is generally not the exact union probability because outcomes with multiple successes are counted repeatedly. Another mistake is to use the independent formula after identifying a shared risk factor. The correction is to model or obtain the joint failure probability, use an explicit shared-condition model, or report bounds rather than implying unsupported precision.
| Available information | Approach | Main qualification |
|---|---|---|
| Marginal probabilities and justified mutual independence | Use one minus the product of failure probabilities. | The independence assumption must apply to the program events jointly. |
| Joint probabilities or a specified joint outcome model | Use the complement of joint failure, or inclusion-exclusion. | Joint inputs must be consistent with the marginals. |
| Marginal probabilities only, without a dependence model | Report the lower and upper bounds. | The exact portfolio probability is not identified by the marginals alone. |
These approaches answer the same probability question with different information. The independent calculation is simple and exact under its assumption; a joint model can represent dependence but needs additional inputs; bounds avoid an unsupported dependence assumption but may be uninformative. The portfolio probability also does not capture how many successes occur, their values, timing, or costs. It is a focused measure of the chance of at least one success, not a complete portfolio evaluation.
Check Your Understanding
Answer each question before reading the solutions.
- Two independent programs have success probabilities \(0.20\) and \(0.50\). What is the probability that at least one succeeds?
- Two programs have marginal success probabilities \(0.35\) and \(0.25\), with joint success probability \(0.10\). Calculate the probability that at least one succeeds.
- Three programs have marginal success probabilities \(0.20\), \(0.30\), and \(0.45\), with no dependence information. Give valid bounds for the probability of at least one success.
- Why can a shared condition make the independent calculation misleading even if each program’s marginal probability is correct?
Solutions. (1) By independence, \(1-(1-0.20)(1-0.50)=1-(0.80)(0.50)=0.60\), so the probability is \(60.0\%\). (2) By inclusion-exclusion, \(0.35+0.25-0.10=0.50\), or \(50.0\%\). (3) The lower bound is the largest marginal, \(0.45\), and the upper bound is the sum, \(0.20+0.30+0.45=0.95\); hence \(0.45\leq P(\text{at least one success})\leq0.95\). (4) Correct marginal probabilities do not specify how program outcomes occur jointly. If a shared condition makes successes cluster in the same scenarios, the probability of all programs failing can differ from the product of their marginal failure probabilities, so the independent formula need not give the correct portfolio probability.