I really need help I do not understand how to start this problem.
A machine that manufactures automobile parts produces defective parts 15% of the time. If 7 parts produced by this machine are randomly selected, what is the probability that at least 3 of the parts are defective? Thank you
To find the probability that at least 3 of the 7 selected parts are defective, you can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = C(n, 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 in a single trial, and
- C(n, k) is the combination formula, given by: C(n, k) = n! / (k! * (n-k)!)
In this case:
- k represents the number of defective parts,
- n is the number of parts selected (7), and
- p is the probability of a part being defective (15%).
To find the probability of at least 3 defective parts, you need to calculate the probability of 3, 4, 5, 6, and 7 defective parts separately and then sum them up.
Let's calculate each probability step-by-step:
P(X = 3):
C(7, 3) * (0.15)^3 * (0.85)^(7-3) = (7! / (3! * (7-3)!)) * (0.15)^3 * (0.85)^4
P(X = 4):
C(7, 4) * (0.15)^4 * (0.85)^(7-4) = (7! / (4! * (7-4)!)) * (0.15)^4 * (0.85)^3
P(X = 5):
C(7, 5) * (0.15)^5 * (0.85)^(7-5) = (7! / (5! * (7-5)!)) * (0.15)^5 * (0.85)^2
P(X = 6):
C(7, 6) * (0.15)^6 * (0.85)^(7-6) = (7! / (6! * (7-6)!)) * (0.15)^6 * (0.85)^1
P(X = 7):
C(7, 7) * (0.15)^7 * (0.85)^(7-7) = (7! / (7! * (7-7)!)) * (0.15)^7 * (0.85)^0
Finally, sum up these individual probabilities to get the probability of at least 3 defective parts:
P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
To solve this problem, we can use the concept of binomial probability. The binomial probability formula is used to calculate the probability of obtaining a specific number of successes in a fixed number of independent Bernoulli trials (in this case, selecting parts).
Let's break down the problem step by step:
Step 1: Determine the relevant parameters:
- Probability of success (p): This refers to the probability of selecting a defective part. In this case, p = 15% = 0.15.
- Number of trials (n): This represents the number of parts selected. Here, n = 7.
- Number of desired successes (k): This refers to the minimum number of defective parts we want. In this case, k = 3 or more (i.e., at least 3).
Step 2: Calculate the probability of exactly k successes:
The probability of exactly k successes (P(X = k)) can be calculated using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Here, (nCk) represents the number of combinations or ways to choose k successes out of n trials. It can be calculated using the formula:
(nCk) = n! / (k! * (n-k)!)
Step 3: Calculate the probability of at least k successes:
To find the probability of at least k successes (P(X >= k)), we need to sum up the probabilities of having exactly k, k+1, k+2, ..., n successes.
P(X >= k) = P(X = k) + P(X = k+1) + P(X = k+2) + ... + P(X = n)
Step 4: Calculate the desired probability:
Using the formulas and steps described above, we can calculate the probability of at least 3 defective parts.
P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
Let's calculate the values and sum them up:
P(X >= 3) = (7C3) * (0.15)^3 * (0.85)^(7-3) +
(7C4) * (0.15)^4 * (0.85)^(7-4) +
(7C5) * (0.15)^5 * (0.85)^(7-5) +
(7C6) * (0.15)^6 * (0.85)^(7-6) +
(7C7) * (0.15)^7 * (0.85)^(7-7)
Now, you just need to calculate these terms and add them up to find the answer.