For a PERT activity with optimistic (to), most likely (tm), and pessimistic (tp) time estimates, the expected time t is computed as t = (to + 4*tm + tp) / 6.

Introduction / Context:
PERT (Program Evaluation and Review Technique) models activity duration uncertainty using three-point estimates: optimistic (to), most likely (tm), and pessimistic (tp). The question asks for the expected time formula used to compute a weighted mean for scheduling and risk-aware planning.

Given Data / Assumptions:

• Three estimates are available: to, tm, tp.
• We need the expected time t in PERT.
• Assume standard Beta-distribution-based weighting used in classical PERT.

Concept / Approach:
Classical PERT applies a weighted average emphasizing the most likely duration. The standard weighting is: to (weight 1), tm (weight 4), tp (weight 1). The denominator 6 normalizes the weights. This prioritizes tm while accounting for the range from best-case to worst-case.

Step-by-Step Solution:
1) Recall PERT mean formula: expected t = (to + 4*tm + tp) / 6.2) Substitute if numerical values are given (not required here).3) Use t for forward pass (earliest times) and for summary statistics of activity duration.

Verification / Alternative check:
Check units: all terms are time; linear combination over 6 maintains dimension of time. Compare with simple average (to + tm + tp)/3, which would underweight the most likely estimate relative to PERT practice.

Why Other Options Are Wrong:

• t = (to + tm + tp) / 3: Simple mean, not PERT's weighted mean.
• t = (4*to + tm + tp) / 6: Overweights optimistic time incorrectly.
• t = (to + 2*tm + 3*tp) / 6: Skews toward pessimistic time, not standard PERT.

Common Pitfalls:

• Forgetting the weight 4 on tm and denominator 6.
• Using PERT mean without assessing variance; remember sigma^2 = ((tp - to) / 6)^2 for classical PERT.

Final Answer:
t = (to + 4*tm + tp) / 6.