Problems: Binomial
(Some of these problems require the use of binomial
tables. If you do not have access to binomial tables, you
can answer these questions using a binomial calculator, such
as this one found on the Internet: http://www.stat.tamu.edu/~west/applets/binomialdemo.html.)
1. Below is a table showing the possible arrangements of
the letters ABC:
AA BA CA
AB BB CB
AC BC CC
 a) Which of these pairs would be counted as
permutations of the three letters taken two at a
time?
 b) How many combinations of three letters taken two
at a time are here?
2. A multiplechoice test has 16 questions on it, and
each question has five possible answers.
 a) If a person guessed at random, what is the
probability that he or she would get exactly 6 right?
(You can use binomial tables to answer this question.)
 b) The teacher who is giving the test is convinced
that John knows absolutely nothing and just guesses
randomly on the test. The teacher wants to test this
hypothesis that John is randomly guessing. Because he
wants convincing evidence that this hypothesis is wrong
before he abandons it, he sets the alpha level at .01.
How many questions would John have to get right before
the teacher would conclude he was not a complete
knucklehead? (Use a onetailed test, and use the binomial
tables. Explain carefully what you are doing.)
3. A multiplechoice test has 16 questions on it, and
each question has four possible answers. If a person guessed
at random, what is the probability that he or she would get
exactly 6 right? (Do not use the tableshow what it looks
like in terms of the formula. Just set it updo not bother
with all the multiplication and division.)
4. Suppose 30% of the students on campus are female. If
ten names are selected at random from the student roster,
what is the probability that five of the names will belong
to female students?
5. Suppose a multiplechoice test has ten questions and
each question has four possible answers.
 a) If a student guesses randomly, what is the
probability that he will get exactly one correct?
 b) What is the average score that such a student can
expect to get?
6. Peter Potts is an avid gardener. In March every year
he starts tomato plants indoors. If he plants exactly one
seed in a dozen different pots, and if the germination rate
for the seeds he has is 90%, what is the probability that
exactly ten of the pots he plants will grow a tomato
plant?
7. Mighty Casey is a baseball player who gets a hit 1/3
of the time he is at bat. Assuming independence, what is the
probability that in today's game Casey will get two hits if
he has four at bats?
8.The Glenville Gledes baseball team has 15 members.
 a) In how many ways can 9 players be selected from
it? (How many teams could be fielded?) Explain how you
get your answer. You do not have to multiply and divide
outjust show what you would multiply and divide.
 b) Only three can pitch, so one of these three must
be selected as the pitcher. In how many ways can 9
players no be selected?
 c) Given that 9 players have been selected, how many
different batting orders are possible?
If in the last six months 60% of all stocks listed on the
New York Stock exchange have fallen in value, what is the
probability that of six stocks picked at random, five will
have declined in price?
9. A Russian Roulette player has played his game eight
times and is still alive.
 a) If the probability of losing is 1/6, find the
exact probability of winning this game eight times
straight.
 b) How many times do you have to play this game
before the probability of losing once are greater than
50%
 c) If eight people play this game one time each at a
party, what is the probability that either one or two
will lose.
 d) Can you use the normal curve approximation to
answer b?
10. A group of college students is making obscene phone
calls. They are dialing numbers randomly and the probability
that someone answers the phone on any given call is .6. If
they make ten calls, what is the probability that exactly
seven people will answer the call?
Even more (I wrote a lot of these in my early years of
teaching statistics):
11. A deck has 52 cards. How many groups of five cards
can be selected from it? How many groups of five can be
selected so there is only one club? Given that five cards
have been selected, in how many ways can they be
ordered?
12. A salesman has found that his sales pitch is
successful 25% of the time. What is the probability that he
will make 4 sales to the next 10 customers?
13. A probabilityminded traveling salesman figures the
probability of making a sale is .2 at each call.
 a) If he is right, what is the exact probability that
he will make either four or five sales today if he makes
eight calls?
 b) Using a normal distribution approximation, answer
part a.
 c) Considering each day as an independent sample,
what is the probability that in five weeks (25 days) he
will average at least three sales a day (assuming he
makes exactly eight calls a day)? Ignore the continuity
correction.
14. Consumer Reports evaluated low price compact
stereos. On one brand they found that two of the three sets
tested had poor stereo separation (cross talk) as the result
of poorly aligned FM circuits.
 a) If only one in ten stereos of this brand actually
have excessive cross talk, what are the odds that of
three selected at random, two would be found with this
defect?
 b) If 50% of the stereos produced by this company
have excessive cross talk, what is the probability that
two of the three selected at random will have this
defect?
15. If a woman is selected randomly from all those who
have five children. What is the probability she will have
four boys and one girl?
16. A teacher has given a test with 18 multiplechoice
questions. Each question has 3 possible answers. If the
teacher insists that the C range begins at least one
standard deviation from the score one could expect from
random guessing, where does the C range begin?
17. A multiplechoice test has 48 questions and each
question has 4 possible answers.
 a) What is the exact probability of getting 15
correct if one guesses randomly?
 b) Work this problem using the normal curve
approximation.
18. Suppose a multiplechoice test has 10 questions and
each question has four answers.
 a) If a student guesses randomly, what is the
probability that he will get exactly two correct?
 b) What is the average score that such a student can
expect to get?
19. Suppose a student takes a multiplechoice test that
has 12 questions and each question has 4 possible answers.
Suppose further that he guesses randomly on the test.
 a) What is the expected value of this
distribution?
 b) What is standard deviation of this
distribution?
 c) What is the probability that he will get a C or
better if he needs to answer at least 6 correctly to get
a C? (Use the normal curve approximation.)
