bottles of thesame sizes produced in a factory are placed in boxes. each box contains 10 bottles. if 8% of the bottles produced are defective, find the probability that a box chosen at random contains at least three defective
To find the probability that a randomly chosen box contains at least three defective bottles, we can use the concept of binomial distribution.
First, let's calculate the probability of having exactly three defective bottles in a single box.
The probability of a bottle being defective is 8%, which means the probability of it being non-defective is 100% - 8% = 92%. Since there are 10 bottles in a box, we need to calculate the probability of having exactly three defective bottles and seven non-defective bottles.
Using the binomial probability formula, the probability of having exactly three defective bottles in a box is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k),
where:
- n = total number of trials (number of bottles in a box) = 10,
- k = number of successes (number of defective bottles) = 3,
- p = probability of success (probability of a bottle being defective) = 8% or 0.08.
Substituting the values into the formula:
P(X = 3) = (10 C 3) * (0.08^3) * (0.92^(10 - 3)).
Use the combination formula:
(10 C 3) = 10! / (3! * (10 - 3)!) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.
Now, substitute the values into the probability formula:
P(X = 3) = 120 * (0.08^3) * (0.92^7).
Next, let's calculate the probability of having four defective bottles, five defective bottles, six defective bottles, seven defective bottles, eight defective bottles, nine defective bottles, and ten defective bottles by following the same procedure.
Once you have calculated the individual probabilities for each scenario, add all the probabilities together to find the probability of having at least three defective bottles in a randomly chosen box.
For example:
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).
By summing up these probabilities, you will get the desired result.