Three percent of the widgets produced by machine 1121 are defective. what is the probablity that a box of 30 widgets produced by that machine contains two or more defective widgets?
To find the probability that a box of 30 widgets produced by machine 1121 contains two or more defective widgets, we need to use the binomial probability formula.
The binomial probability formula calculates the probability of getting a certain number of successes (defective widgets) in a fixed number of trials (widgets in the box) when the probability of success (defective widget) remains constant.
The formula is as follows:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of getting exactly x successes,
C(n, x) is the number of combinations of n items taken x at a time,
p is the probability of a success,
n is the total number of trials.
In this case, the probability of a defective widget is 3% or 0.03, and we want to find the probability of having two or more defective widgets in a box of 30 widgets.
To calculate this, we need to calculate the individual probabilities of having exactly two, three, four, and so on, up to 30 defective widgets. Then, we add up all these probabilities to get the final answer.
P(2 or more defective widgets) = P(2) + P(3) + P(4) + ... + P(30)
Let's calculate the probability for each scenario and then sum them up:
P(2) = C(30, 2) * 0.03^2 * (1 - 0.03)^(30 - 2)
P(3) = C(30, 3) * 0.03^3 * (1 - 0.03)^(30 - 3)
P(4) = C(30, 4) * 0.03^4 * (1 - 0.03)^(30 - 4)
...
P(30) = C(30, 30) * 0.03^30 * (1 - 0.03)^(30 - 30)
Finally, we add up all these probabilities to get the final answer.
P(2 or more defective widgets) = P(2) + P(3) + P(4) + ... + P(30)
You can calculate each of these individual probabilities using a scientific calculator or spreadsheet software. Once you have obtained all the probabilities, simply add them up to get the final result.