Detailed Solution
1. Identify Individual Probabilities
For any single person (Ai) in the group:
Probability of dying before 90, P(D)=1/3.
Probability of surviving until 90, P(S)=1−1/3=2/3.
2. Understanding "First to Die"There are several mutually exclusive scenarios where A1 is the first to die among the four persons before they reach age 90:
Scenario 1: Exactly 1 person dies (and it must be A1)
A1 dies, and A2,A3,A4 all survive.
Probability: (1/3)×(2/3)×(2/3)×(2/3)=8/81.
Scenario 2: Exactly 2 persons die (and A1 dies first)
If 2 people die, the probability that A1 is among them is (13) ways to pick the other person.
The probability that A1 dies before the other person is 1/2.
Probability: (13)×(1/3)2×(2/3)2×(1/2)=3×(1/9)×(4/9)×(1/2)=6/81.
Scenario 3: Exactly 3 persons die (and A1 dies first)
Probability that A1 is among 3 people dying is (23) ways to pick the others.
The probability that A1 is the first among the 3 is 1/3.
Probability: (23)×(1/3)3×(2/3)1×(1/3)=3×(1/27)×(2/3)×(1/3)=2/81.
Scenario 4: All 4 persons die (and A1 dies first)
All 4 die, and the probability A1 is the first is 1/4.
Probability: (1/3)4×(1/4)=(1/81)×(1/4)=1/324.
3. Total ProbabilityTo find the total probability, we sum these cases:
P=818+816+812+3241
P=8116+3241
Convert to a common denominator (324=81×4):
P=32416×4+3241=32464+1=32465
