Suppose that every day there is a 1% chance of the power lines failing. What is the percentage chance that they never fail for a whole 3 years?

Steps to Solve

1. Identify the daily failure probability The problem states that there is a 1% chance of failure each day. This means: $$ P(\text{failure}) = 0.01 $$

2. Calculate the daily success probability The probability that the power line does NOT fail in a day is: $$ P(\text{no failure}) = 1 - P(\text{failure}) = 1 - 0.01 = 0.99 $$

3. Determine the total number of days in 3 years Calculate the total number of days in 3 years, assuming a regular year of 365 days: $$ \text{Total days} = 3 \times 365 = 1095 \text{ days} $$

4. Calculate the probability of never failing over 3 years The probability of the power line never failing over 3 years is the success probability raised to the power of the number of days: $$ P(\text{never fails}) = (P(\text{no failure}))^{\text{Total days}} = (0.99)^{1095} $$

5. Calculate the final probability Using a calculator to find: $$ (0.99)^{1095} \approx 0.0000423 $$

6. Convert to percentage To express this probability as a percentage: $$ \text{Percentage} = 0.0000423 \times 100 \approx 0.00423% $$