A factory finds out that on the average 20% of bolts produced by the given machine will be defective.if 10 bolts are selected at random from the day's production of this machine,find the probability that 1.exactly 2 will be defective.

2. 2 or more will be defective
3. More than 5 will be defective

20

To answer these questions, we need to use the binomial probability formula. The binomial probability formula can be used when we have a fixed number of trials, each with two possible outcomes (success or failure), and the probability of success remains constant for each trial. In this case, the trials are the selection of bolts, the outcomes are whether a bolt is defective or not, and the probability of success (getting a defective bolt) is 20%.

The formula for the binomial probability is:

P(x) = (nCx) * p^x * q^(n-x)

Where:
P(x) is the probability of getting exactly x successes,
n is the total number of trials,
x is the number of successes,
p is the probability of success,
q is the probability of failure (1 - p), and
(nCx) is the number of ways to choose x items out of n.

Now let's calculate the probabilities for the given questions:

1. Exactly 2 will be defective:
n = 10 (total number of bolts selected)
x = 2 (number of defective bolts)
p = 0.2 (probability of a bolt being defective)
q = 1 - p = 1 - 0.2 = 0.8 (probability of a bolt not being defective)

P(2) = (10C2) * (0.2)^2 * (0.8)^8
= (10! / (2!(10-2)!)) * (0.2)^2 * (0.8)^8
= (10! / (2!8!)) * 0.04 * 0.16777216
= (10 * 9 / 2) * 0.04 * 0.16777216
≈ 0.301989888

Therefore, the probability that exactly 2 bolts will be defective is approximately 0.30199.

2. 2 or more will be defective:
P(2 or more) = P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)

To calculate the probabilities for each case, you can follow the same steps as in question 1, but substitute x with the respective number of successes and add them up.

3. More than 5 will be defective:
P(more than 5) = P(6) + P(7) + P(8) + P(9) + P(10)

Again, follow the steps as in question 1, but calculate the probabilities for each case and add them up.

Remember to use a calculator or excel sheet to make the calculations easier.