Suppose 5% of chips manufactured at a plant are defective. How many should he/she buy
in order that there is more than 99% chance of having at most one defective chip?

We are dealing with a binomial probability problem where we want to determine how many chips to buy so that there is more than a 99% chance of having at most one defective chip. 

Let:
- \( p = 0.05 \) (probability of a chip being defective),
- \( n \) be the number of chips bought,
- \( X \) be the random variable representing the number of defective chips, and
- \( X \sim \text{Binomial}(n, 0.05) \).

We want \( P(X \leq 1) > 0.99 \).

### Step 1: Calculate the binomial probabilities

For \( P(X \leq 1) \), we need:
\[
P(X \leq 1) = P(X = 0) + P(X = 1)
\]
where \( X \) follows a binomial distribution:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Thus:
\[
P(X = 0) = (1 - p)^n = (0.95)^n
\]
\[
P(X = 1) = \binom{n}{1} p (1 - p)^{n-1} = n \cdot 0.05 \cdot (0.95)^{n-1}
\]

### Step 2: Solve the inequality

We need to find the smallest \( n \) such that:
\[
P(X = 0) + P(X = 1) > 0.99
\]

This requires solving:
\[
(0.95)^n + n \cdot 0.05 \cdot (0.95)^{n-1} > 0.99
\]

I will now compute the value of \( n \) that satisfies this inequality.It turns out that buying just **1 chip** will give a more than 99% chance of having at most one defective chip. This is because the probability of even one chip being defective in such a small sample is extremely low given the 5% defect rate.

Would you like further details or explanations? 

Here are 5 related questions you might want to explore:
1. How would the result change if the defect rate was higher?
2. What if I want a 99.9% chance instead of 99%?
3. How would you approach the problem if you wanted at most 2 defective chips?
4. Can you explain the binomial distribution further?
5. How would this change if chips were bought in batches?

**Tip:** The binomial distribution is ideal for scenarios where each trial is independent with only two possible outcomes (success or failure).

Suppose 5% of chips manufactured at a plant are defective. How many should he/she buy in order that there is more than 99% chance of having at most one defective chip?

Solve a binomial probability problem to determine how many chips to buy with a 5% defect rate and over 99% chance of having at most one defective chip. The solution covers key formulas and concepts such as binomial distribution and inequalities. Suitable for Grades 11-12.

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