Suppose 2% of the items made in a factory are defected. Find the probability that no defective in a sample of 50.
To find the probability that there are no defective items in a sample of 50 items, we can use the binomial distribution. Let's break down the steps to calculate this probability:
Step 1: Understand the Problem
From the problem statement, we know that 2% of the items made in the factory are defective. Therefore, the probability of any individual item being defective is 0.02 (or 2%).
Step 2: Define the Variables
Let's define the variables we need for our calculation:
- p: The probability of an item being defective (p = 0.02).
- n: The total number of items in the sample (n = 50).
- X: The number of defective items in the sample.
Step 3: Apply the Binomial Probability Formula
The probability of getting exactly X defective items in a sample of size n can be calculated using the binomial probability formula:
P(X) = (n C X) * p^X * (1-p)^(n-X)
In this case, we want to find the probability of getting zero defective items, so X = 0.
P(X = 0) = (n C 0) * p^0 * (1-p)^(n-0)
= (n C 0) * 1 * (1-p)^n
= 1 * 1 * (1-p)^n
= (1-p)^n
Step 4: Plug in the Values and Calculate
Now, we can substitute the values into the formula to find the probability:
P(X = 0) = (1 - 0.02)^50
= 0.98^50
≈ 0.366032
Therefore, the probability of having no defective items in a sample of 50 from this factory is approximately 0.366 (or 36.6%).