n independent men, each of age x years. For a given man, the probability of dying within a years is p. What is the probability that a particular man M_k is the first to die (symmetry assumption for ties)?

Introduction / Context:
Each of n men independently dies within a years with probability p and survives that period with probability (1 − p). We want the probability that a particular individual M_k is the first to die. Because the model is discrete over a fixed period, “first” is interpreted via symmetry: if at least one death occurs, all n individuals are equally likely to be the one identified as “first.”

Given Data / Assumptions:

• Independence across individuals.
• P(death within a years) = p; P(survival) = 1 − p.
• At least one death event in the cohort has probability 1 − (1 − p)^n.
• Ties (multiple deaths) are assigned uniformly by symmetry as per standard exam convention.

Concept / Approach:
Condition on the event “there is at least one death.” Under symmetry among the n men, the chance that M_k is selected as the first equals 1/n given that at least one death happens. Unconditioning multiplies by the probability that at least one death occurs.

Step-by-Step Solution:
P(at least one death) = 1 − (1 − p)^n.By symmetry, P(M_k is the identified first | at least one death) = 1/n.Therefore P(M_k is first) = (1/n) × [1 − (1 − p)^n].

Verification / Alternative check:
The n such probabilities sum to 1 − (1 − p)^n across all k, matching the total probability that someone dies, as required.

Why Other Options Are Wrong:
1 − (1 − p)^n is the probability that at least one death occurs (not tied to a specific person); 1/n^2 variants are dimensionally incorrect; p(1 − p)^{n−1} would be the probability that exactly one specific person dies and all others survive, not accounting for scenarios where multiple die.

Common Pitfalls:
Ignoring the conditional symmetry or misinterpreting “first” in a discrete-period setup. The adopted convention aligns with many exam problems.

Final Answer:
1/n[1 - (1 - p)^n]