Transcript Sets
Assignment 2
Due 16 December 2009, Wednesday
Please hand in to the Resources Office
Binomial Experiment
Experiment of m repeated trials where the following are valid:
1. all trials are statistically independent (in the sense that
knowing the outcome of any particular one of them does not
change one’s assessment of chance related to any others);
2. each trial results in only one of two possible outcomes,
labeled as “success” and “failure”;
3. the probability of success on each trial, denoted by p,
remains constant.
Binomial Random Variable
X = Number of trials that result in a success
X is said to have Binomial distribution with
parameters m and p.
Binomial PMF
p (x) =
m x
m x
p (1 p)
x
where
m
m!
x x!(m x)!
Factorial
m! = m x (m – 1) x …. x 2 x 1
0! = 1
1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
Expectation
E(X) = m p
Variance
Var (X) = m p (1 – p)
Example 1
Two building contractors, A and B, have
competed for 50 contracts. A won 20 and B
won 30. The contractors asked to tender for
3 new contracts. What is the probability that
A) Contractor A will win all the contracts
(0.064);
B) Contractor B will win at least one contract
(0.936);
C) Contractor A will win exactly two contracts
(0.288)?
Example 2
A food-packaging apparatus under-fills 10% of
the containers. Find the probability that for
any particular 5 containers the number of
under-filled will be:
A) exactly 3 (0.0081);
B) exactly 2 (0.0729);
C) zero (0.590);
D) at least 1 (0.41).
Example 3
A particular jury consists of 7 jurors. Each juror
has a 0.2 chance of making the wrong
decision, independently of the others. If the
jury reaches a decision by majority rule,
what is the probability that it will reach a
wrong decision? (0.0333)
Example 4
In a shotgun seminar, the topic is announced 1 week
in advance, but the speaker is randomly chosen
when the seminar begins. Suppose that you
attend 12 seminars with 19 other participants.
(i) What is the expected number of times you will be
selected as speaker? (0.6)
(ii) What is the probability that you will be selected
exactly twice? (0.099)
(iii) Suppose that the 20 participants are divided into
five teams of four for preparation. What is the
probability that your team will be selected exactly
twice? (0.283)