DTM.3:5.4 - Repeated opportunities do not create repeated first uptake

For one initially eligible recipient, suppose two independent comparable opportunities each succeed with probability 0.8. The probability of first uptake during those opportunities is 1−0.2²=0.96. It can also be obtained as 0.8+0.2×0.8: the second term includes only the recipients still eligible after the first opportunity.

The product 2×0.8=1.6 instead gives the expected number of successes when both attempts are actually performed and success is repeatable. It is not a first-uptake probability. Either quantity can be useful, but they feed different receiving models. The participant-state coupling must preserve that distinction even when the same internal readiness supplies both probabilities.