I'm stuck on this homework question. If someone can just help get me started with the first one I'm sure I can do the rest. I'm just not sure how to begin. Thanks!

1. A production process produces 90% non-defective parts. A sample of 10 parts from the production process is selected.
a. What is the probability that the sample will contain 7 non-defective parts?
b. What is the probability that the sample will contain at least 4 defective parts?
c. What is the probability that the sample will contain less than 5 non-defective parts?
d. What is the probability that the sample will contain no defective parts?

how did you do them? do you have an equation or anything?

To solve these probability problems, we can use the binomial probability formula. The binomial probability formula is given by:

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

Where:
- n is the number of trials or the sample size
- k is the number of successful outcomes
- p is the probability of success in one trial

Now let's apply this formula to solve each part of the question.

a. To find the probability that the sample will contain 7 non-defective parts, we use the binomial probability formula with n = 10 (sample size), k = 7 (number of successful outcomes), and p = 0.9 (probability of success):

P(X = 7) = (10 C 7) * 0.9^7 * (1-0.9)^(10-7)

b. To find the probability that the sample will contain at least 4 defective parts, we need to find the probabilities of getting 4, 5, 6, 7, 8, 9, and 10 defective parts and add them together.

P(X >= 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

c. To find the probability that the sample will contain less than 5 non-defective parts, we need to find the probabilities of getting 0, 1, 2, 3, and 4 non-defective parts and add them together.

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

d. To find the probability that the sample will contain no defective parts, we use the binomial probability formula with n = 10 (sample size), k = 0 (number of successful outcomes), and p = 0.9 (probability of success).

To solve these probability questions, you can make use of the binomial probability formula. The binomial probability formula calculates the probability of obtaining a specific number of successes in a fixed number of independent Bernoulli trials.

The formula for the binomial probability is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k),

where P(X=k) is the probability of getting exactly k successes, n is the number of trials, p is the probability of success on a single trial, and (n choose k) is the binomial coefficient representing the number of different combinations of k successes.

Let's break down each part of the question and solve them step by step:

a. What is the probability that the sample will contain 7 non-defective parts?

Using the binomial probability formula, where n = 10 (sample size), k = 7 (number of non-defective parts), and p = 0.9 (probability of a non-defective part), we can calculate:

P(X=7) = (10 choose 7) * (0.9)^7 * (1-0.9)^(10-7)

The binomial coefficient, (10 choose 7), can be calculated as follows:
(10 choose 7) = 10! / (7! * (10-7)!) = 10! / (7! * 3!) = (10*9*8) / (3*2*1) = 120

Substituting these values into the formula:
P(X=7) = 120 * (0.9)^7 * (0.1)^3

You can calculate the final result using a calculator.

b. What is the probability that the sample will contain at least 4 defective parts?

To calculate the probability of at least 4 defective parts, you need to calculate the probability of getting exactly 4 defective parts, then exactly 5, 6, 7, 8, 9, and 10 defective parts. Then, you sum up these probabilities.

For each case, you will use the binomial probability formula as mentioned above, with n = 10 (sample size), k (number of defective parts) varying from 4 to 10, and p = 0.1 (probability of a defective part). Calculate each of these probabilities and sum them to get the final result.

c. What is the probability that the sample will contain less than 5 non-defective parts?

To calculate the probability of less than 5 non-defective parts, you need to calculate the probabilities of getting exactly 0, 1, 2, 3, and 4 non-defective parts. Then, you sum up these probabilities.

Using the binomial probability formula, where n = 10 (sample size), k (number of non-defective parts) varying from 0 to 4, and p = 0.9 (probability of a non-defective part), calculate each of these probabilities and sum them to get the final result.

d. What is the probability that the sample will contain no defective parts?

This can be calculated using the binomial formula with n = 10, k = 0, and p = 0.1. Calculate P(X=0) using the formula mentioned earlier.

Remember to substitute the values and perform the necessary calculations using a calculator or statistical software.

I hope this explanation helps you get started on your homework! Let me know if you need any further assistance with the specific calculations.

a. 3/4

b. 4/8
c. 1/4
d. 2/7