91Ó°ÊÓ

In the context of this problem, the probability of failure is a crucial concept. It is defined by the probability that a single event does not succeed. In our exercise, this event is the inability of a camera to pass the test. Given that a camera has a 0.8 probability of passing, the probability it fails is simply:

• Failure Probability = 1 - Passing Probability
• Failure Probability = 1 - 0.8 = 0.2

Thus, the probability of failure for each camera is 0.2, implying a 20% chance that an individual camera won't meet the criteria. Understanding this helps set the stage for upcoming calculations involving groups of cameras.

Binomial probability is used when calculating events with two possible outcomes: success or failure. Each camera being tested represents a trial, where the outcome is either it passes or it does not. The concept helps us understand how likely it is for at least one camera to fail out of a group.In our scenario, we consider the binomial distribution of n cameras all passing the test. The probability of this is given by raising the success probability (0.8) to the power of n: \[ \text{Probability of all n cameras passing} = 0.8^n \]The complementary probability, that at least one camera fails, is calculated as:\[ 1 - 0.8^n \]This formula is essential to identify the smallest n such that this complementary probability reaches or exceeds 95%.

Inequality solving involves finding the values of variables that satisfy a given inequality condition. In this exercise, we set the condition that the probability of at least one camera failing should be no less than 95%. Hence, we have the inequality:\[ 1 - 0.8^n \geq 0.95 \]By solving this inequality, we rearrange it to:\[ 0.8^n \leq 0.05 \]This shows us the need to solve for n, which represents the sample size. Solving inequalities often requires manipulations like taking natural logs. This allows us to linearize the problem, making it easier to handle.

The natural logarithm, often denoted as \( \ln \), is a logarithm to the base e, where e is approximately 2.71828. It is particularly helpful in simplifying calculations involving exponential growth or decay. In the given problem, the natural logarithm is employed to solve the inequality by converting an exponential function into a linear relationship.After establishing the inequality:\[ 0.8^n \leq 0.05 \]We take the natural logarithm of both sides to remove the exponent, which results in:\[ n \cdot \ln(0.8) \leq \ln(0.05) \]This transformation changes multiplicative relationships into additive ones, easing the calculation process. Since \( \ln(0.8) \) is negative, dividing both sides by it reverses the inequality sign:\[ n \geq \frac{\ln(0.05)}{\ln(0.8)} \]This step is vital for determining the smallest integer n, representing the required sample size.