91Ó°ÊÓ

The complement rule is a fundamental concept in probability theory.
It helps us to find the probability of an event by knowing the probability of its complement.

The complement of an event is essentially everything that is not part of the event itself.
For example, if event A is 'getting at least one defective iPhone', then its complement, denoted as \(\bar{A} \), is 'not getting any defective iPhones'.

The sum of the probabilities of an event and its complement always equals 1.
This can be mathematically represented as:

\[ P(A) + P(\bar{A}) = 1 \]

To find the probability of event A, if we know the probability of its complement \(\bar{A} \), we can use:

\[ P(A) = 1 - P(\bar{A}) \]

In our exercise, we wanted to find the probability of getting at least one defective iPhone (event A).
By calculating the probability of picking three good iPhones in a row (complement of A),
we could then use the complement rule to find our desired probability:
\[ P(A) = 1 - P(\bar{A}) = 1 - 0.857 = 0.143 \]

Understanding defective probability is important for solving this problem.
Given a batch of iPhones where 5% are defective, we denote the probability of selecting a defective iPhone as 0.05.

Conversely, this means the probability of selecting a good iPhone is 0.95.
In our scenario, we select 3 iPhones with replacement, meaning each selection is independent of the others.

When dealing with multiple selections, we use the product rule to calculate the combined probability of sequences of events.
For example, the probability of selecting 3 good iPhones in a row is:
\[ (0.95) \times (0.95) \times (0.95) = 0.857 \]

This calculation helped us find the complement probability in our problem.

The probability of events involves determining how likely a particular outcome is.
In our exercise, we were interested in the event A: 'getting at least one defective iPhone'.

Probability theory breaks down such problems step-by-step:

• Identify the individual probabilities of basic events (e.g., selecting a good or defective iPhone).
• Combine these using rules of probability (addition, multiplication, and complement rules).

By defining our events and their complements, we simplified our calculations.

Notice how we evaluated the given options:

• Option a correctly calculates \( P(\bar{A}) = 0.857 \).
• Option b correctly uses the complement rule to find \( P(A) = 0.143 \).
• Option c mistakenly tries to calculate \( P(A) \) by multiplying the defective probabilities directly, which is incorrect in this context.

This methodical approach ensures clarity when solving probability problems.