A machine has a daily failure probability of 0.95 (so its daily working probability is 0.05). What is the probability that it works for four consecutive days without failing?

Introduction / Context:
This question tests independent-trial probability. If a device has a given chance to function (or fail) each day, the probability of a multi-day run is the product of day-wise probabilities, assuming independence from day to day.

Given Data / Assumptions:

• Daily failure probability = 0.95.
• Hence daily working probability = 1 − 0.95 = 0.05.
• Days are independent (standard assumption in such problems).
• We require “works for four consecutive days”.

Concept / Approach:
For independent events A, B, C, D, P(A∩B∩C∩D) = P(A) * P(B) * P(C) * P(D). Here, each day has the same success probability p = 0.05.

Step-by-Step Solution:

Verification / Alternative check:
Express 0.05 as 5 × 10^−2. Then (5 × 10^−2)^4 = 5^4 × 10^−8 = 625 × 10^−8 = 6.25 × 10^−6 = 0.00000625. Matches the direct computation.

Why Other Options Are Wrong:
0.0625 is (0.25)^2, unrelated here; 0.16548375 and 0.000625 come from incorrect powers or mixing failure vs success probability; “None of these” is not needed because the exact value appears among the options.

Common Pitfalls:
Using the failure probability instead of the working probability, or raising to the wrong power (e.g., three days) will yield incorrect values. Independence across days must also be assumed explicitly.

Final Answer:
0.00000625