A lot of 1000 components contains 300 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective.
a. Find P(A).
b. Find P(B|A).
c. Find P(A ? B).
d. Find 
.
e. Find P(B).
f. Find P(A|B).
g. Are A and B independent? Is it reasonable to treat A and B as though they were independent? Explain.
(a) P(A) = 300?1000 = 3?10
(b) Given that A occurs, there are 999 components remaining, of which 299 are defective.
Therefore P(B|A) = 299?999.
(c) P(A ? B) = P(A)P(B|A) = (3?10)(299?999) = 299?3330
(d) Given that 
occurs, there are 999 components remaining, of which 300 are defective.
Therefore P(B|A) = 300?999. Now 
= (7?10)(300?999) = 70?333.
(e) P(B) = P(A ? B) + 
= 299?3330 + 70?333 = 3?10
(g) A and B are not independent, but they are very nearly independent. To see this note that P(B) = 0.3, while P(B|A) = 0.2993. So P(B) is very nearly equal to P(B|A), but not exactly equal. Alternatively, note that P(A ? B) = 0.0898, while P(A)P(B) = 0.09. Therefore in most situations it would be reasonable to treat A and B as though they were independent.
You might also like to view...
Technician A says that oil pressure is normally higher when the engine is cold. Technician B says that normal engine oil pump pressure ranges from 20 to 80 psi (200 to 400 kPa) or 20 psi for every 1000 engine RPM. Who is right?
A. Both technicians A and B B. Technician B only C. Neither technician A nor B D. Technician A only
E-85 is a mixture of _____percent gasoline and ____ percent ethanol.
Fill in the blank(s) with the appropriate word(s).
Fuses and circuit breakers provide the needed protection for motors
Indicate whether the statement is true or false.
Plants are the basis of the nursery industry
Indicate whether the statement is true or false