But there are 3 6 different orderings so the number

Example 2.11. Continuing from Example 2.10, assume instead of a board we are interested in the
number of possible
committees , where each member has equal power. How many such committees are possible?
20

Application of Combinations to Probability Problems .
Example 2.13. A lot of 100 articles contains 10 defective. If a sample of 5 articles is chose at random,
what is the probability that it contains 3 defective?
Example 2.14. A player serving at tennis is only allowed one fault. At a double fault, the sever loses
a point and the other player gains a point. Given the following information:
21

What is the probability that the server loses a point, i.e., what is the probability that both services
result in faults?
Example 2.15. Referring to Example 2.6. Given that the length of a rod is too long, what is the
probability that the diameter is OK?
Diameter
Length Too Thin OK Too Thick Sum
Too Short 10 3 5 18
OK 38 900 4 942
Too Long 2 25 13 40
Sum 50 928 22 1000
22

2.4.1 Independent Events
When the given occurrence of one event does not influence the probability of a potential outcome of
another event, then the two events are said to be independent.
Two events A and B are independent if the probability of each remains the same, whether or not
the other has occurred. A and B are independent if P ( A ) 6 = 0 and P ( B ) 6 = 0, then
P ( B | A ) = P ( B ) ⇐⇒ P ( A | B ) = P ( B )
If P ( A ) = 0 or P ( B ) = 0, then the two events are independent.
More generally, the events A
1
, ...A
n
are independent if for each A
i
and each collection A
j 1
, ..., A
jm
of
events with P ( A
j 1
∩ · · · ∩ A
jm
) 6 = 0
P ( A
i
| A
j 1
, ..., A
jm
) = P ( A
i
)
As a consequence of independence, the rule of multiplication then says
P ( A ∩ B ) = P ( A | B ) P ( B ) = P ( A ) P ( B )
In general,
P
k \
i =1
A
i
!
=
k Y
i =1
P ( A
i
) k ≥ 1
Example 2.16. Of the microprocessors manufactured by a certain process, 20% of them are defec-
tive. Assume they function independently. Five microprocessors are chosen at random. What is the
probability that they will all work?