Answer:
a. The probability that all 5 selected transistors are defective is 0.0024 or 0.24%.
To calculate this probability, we use the formula for the probability of the intersection of independent events:
P(all 5 defective) = P(defective) x P(defective) x P(defective) x P(defective) x P(defective)
where P(defective) = 5/12 for each transistor.
Plugging in the values, we get:
P(all 5 defective) = (5/12) x (5/12) x (5/12) x (5/12) x (5/12) = 0.0024
b. The probability that none of the selected transistors are defective is 0.163 or 16.3%.
To calculate this probability, we use the complement rule:
P(none defective) = 1 - P(at least one defective)
To calculate P(at least one defective), we can use the complement rule again:
P(at least one defective) = 1 - P(none defective)
So, we need to find P(none defective) first:
P(none defective) = P(not defective) x P(not defective) x P(not defective) x P(not defective) x P(not defective)
where P(not defective) = 7/12 for each transistor.
Plugging in the values, we get:
P(none defective) = (7/12) x (7/12) x (7/12) x (7/12) x (7/12) = 0.163
Then, we can find P(at least one defective):
P(at least one defective) = 1 - P(none defective) = 1 - 0.163 = 0.837
Finally, we can find P(none are defective) using the complement rule:
P(none defective) = 1 - P(at least one defective) = 1 - 0.837 = 0.163.