Step-by-step explanation:
the probabilty for exactly 48 people surviving is
(64 over 48) × 0.8⁴⁸ × 0.2¹⁶
the possible combinations of 48 people out of 64 associated with the probability of 48 survivors and 16 deaths.
"at least" is then sum of the probabilities of exactly 48, 49, 50, 51, ... 64 people surviving.
that is a longer calculation of these 17 possibilities.
they go
(64 over 48) × 0.8⁴⁸ × 0.2¹⁶ = 0.0714...
(64 over 49) × 0.8⁴⁹ × 0.2¹⁵ = 0.0933...
(64 over 50) × 0.8⁵⁰ × 0.2¹⁴ = 0.1119...
...
(64 over 63) × 0.8⁶³ × 0.2¹ = 0.00001004...
(64 over 64) × 0.8⁶⁴ × 0.2⁰ = 0.0000006277...
the sum is the probabilty (that at least 48 out of the 64 are alive at 64) of 0.8746...
the maximum single probability (for an exact number of survivors) is as expected at exactly 51 and exactly 52 survivors with a probability of 0.1229... for each case.
of course, this is around the expectation value of
64×0.8 = 51.2 (out of the 64 it is expected that per average 51.2 people survive and reach age 64).
