Probability of Defective Tools

To find the probability that at most 2 tools will be defective in a sample of 40, we need to calculate the cumulative probability of having 0, 1, or 2 defective tools.

Step 1: Calculate the Probability of a Defective Tool

Given that 5% of the tools produced are defective, the probability of a tool being defective is 0.05. Let's denote this probability as P(defective).

Step 2: Calculate the Probability of a Non-defective Tool

The complement of a defective tool is a non-defective tool. Since there are only two possibilities (defective or non-defective), the probability of a tool being non-defective is 1 - P(defective). Let's denote this probability as P(non-defective).

P(non-defective) = 1 - P(defective) = 1 - 0.05 = 0.95

Step 3: Calculate the Probability of 0 Defective Tools

To calculate the probability of having 0 defective tools in a sample of 40, we need to use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:
- P(X = k) is the probability of getting exactly k successes (in this case, 0 defective tools)
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of a single success (defective tool)
- n is the total number of trials (sample size)
- k is the number of successes (defective tools)

In our case, k = 0, p = 0.05, n = 40. Plugging these values into the formula:

P(X = 0) = C(40, 0) * 0.05^0 * (1 - 0.05)^(40 - 0)

C(40, 0) is equal to 1, and any number raised to the power of 0 is 1. Therefore:

P(X = 0) = 1 * 1 * (1 - 0.05)^40 = 0.95^40

Step 4: Calculate the Probability of 1 Defective Tool

Using the same formula as in step 3, but with k = 1, p = 0.05, and n = 40:

P(X = 1) = C(40, 1) * 0.05^1 * (1 - 0.05)^(40 - 1)

C(40, 1) is equal to 40, so:

P(X = 1) = 40 * 0.05 * (1 - 0.05)^39 = 40 * 0.05 * 0.95^39

Step 5: Calculate the Probability of 2 Defective Tools

Using the same formula as in step 3, but with k = 2, p = 0.05, and n = 40:

P(X = 2) = C(40, 2) * 0.05^2 * (1 - 0.05)^(