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In probability theory, computing the aggregate probability of an event often requires considering all possible scenarios or pathways through which that event can occur. This approach relies on the principle of partition-based probability aggregation, which provides an elegant and systematic way to calculate the likelihood of an event by decomposing the sample space into mutually exclusive and exhaustive subsets.
Suppose we want to determine the probability of an event A occurring, but this probability depends on different underlying conditions or scenarios. If we can partition the sample space into n mutually exclusive events {B₁, B₂, ..., Bₙ} that collectively cover all possibilities (i.e., exactly one of them must occur), then we can express the probability of A as a weighted sum:
$$P(A) = \sum_{i=1}^{n} P(A|B_i) \cdot P(B_i)$$
Where:
Consider a quality control scenario: A manufacturing facility operates three production shifts—morning, afternoon, and night. Each shift handles a known proportion of the daily production volume (prior probabilities), and each shift has a characteristic error rate (conditional probabilities). To compute the overall defect rate across all production, we weight each shift's defect rate by its production share and sum the results.
This same principle applies universally across domains: from computing disease prevalence across multiple patient populations to estimating message delivery success across multiple network routes.
Implement a function that calculates the aggregate probability of an event given:
Return the computed aggregate probability P(A), rounded to 4 decimal places.
priors = {'B1': 0.3, 'B2': 0.7}
conditionals = {'B1': 0.2, 'B2': 0.5}0.41We have two partition events B1 and B2 with prior probabilities 0.3 and 0.7 respectively.
Applying the partition-based aggregation formula:
P(A) = P(A|B1) × P(B1) + P(A|B2) × P(B2) P(A) = (0.2 × 0.3) + (0.5 × 0.7) P(A) = 0.06 + 0.35 P(A) = 0.41
The aggregate probability of event A is 0.41.
priors = {'B1': 0.2, 'B2': 0.3, 'B3': 0.5}
conditionals = {'B1': 0.1, 'B2': 0.4, 'B3': 0.6}0.44We have three partition events with the following prior and conditional probabilities:
• B1: Prior = 0.2, Conditional = 0.1 • B2: Prior = 0.3, Conditional = 0.4 • B3: Prior = 0.5, Conditional = 0.6
Applying the formula:
P(A) = P(A|B1) × P(B1) + P(A|B2) × P(B2) + P(A|B3) × P(B3) P(A) = (0.1 × 0.2) + (0.4 × 0.3) + (0.6 × 0.5) P(A) = 0.02 + 0.12 + 0.30 P(A) = 0.44
The aggregate probability of event A is 0.44.
priors = {'B1': 0.25, 'B2': 0.25, 'B3': 0.25, 'B4': 0.25}
conditionals = {'B1': 0.1, 'B2': 0.2, 'B3': 0.3, 'B4': 0.4}0.25We have four equally likely partition events (uniform distribution with P(Bᵢ) = 0.25 for all i).
When priors are uniform, the aggregate probability simplifies to the arithmetic mean of the conditional probabilities:
P(A) = 0.25 × (0.1 + 0.2 + 0.3 + 0.4) P(A) = 0.25 × 1.0 P(A) = 0.25
Alternatively, computed step by step: P(A) = (0.1 × 0.25) + (0.2 × 0.25) + (0.3 × 0.25) + (0.4 × 0.25) P(A) = 0.025 + 0.05 + 0.075 + 0.1 P(A) = 0.25
The aggregate probability of event A is 0.25.
Constraints