2) A lot consists of 20 defective and 80 non-defective items from which two items are
chosen without replacement. Events A & B are defined as A = {the first item
chosen is defective}, B = {the second item chosen is defective}
a. What is the probability that both items are defective?
b. What is the probability that the second item is defective?
3) In the above example 2, if we choose 3 items after other without replacement, what
is the probability that all items are defective?
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a. The probability that the first item chosen is defective is 20/100 = 1/5.
After the first defective item is chosen, there are 19 defective items and 99 non-defective items left.
So the probability that the second item chosen is defective, given that the first item was defective, is 19/98.
To find the probability that both items are defective, we multiply the probabilities: (1/5) * (19/98) = 19/490.
b. The probability that the second item is defective can be found by considering two scenarios:
Scenario 1: The first item chosen is defective
In this case, there are 19 defective items and 99 non-defective items left, so the probability is 19/118.
Scenario 2: The first item chosen is non-defective
In this case, there are still 20 defective items and 79 non-defective items left, so the probability is 20/99.
To find the overall probability that the second item is defective, we need to consider both scenarios:
Probability = (1/5) * (19/118) + (4/5) * (20/99) = 239/1180.
3) If we choose 3 items after each other without replacement, the probability that all items are defective can be found as follows:
The probability that the first item chosen is defective is 20/100 = 1/5.
After the first defective item is chosen, there are 19 defective items and 99 non-defective items left.
So the probability that the second item chosen is defective, given that the first item was defective, is 19/99.
After the second defective item is chosen, there are 18 defective items and 98 non-defective items left.
So the probability that the third item chosen is defective, given that the first two items were defective, is 18/98.
To find the probability that all three items are defective, we multiply the probabilities: (1/5) * (19/99) * (18/98) = 342/48510.