Start . . Text Back

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 multiple-choice 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 one-tailed test, and use the binomial tables. Explain carefully what you are doing.)

3. A multiple-choice 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 table--show what it looks like in terms of the formula. Just set it up--do 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 multiple-choice 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 out--just 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 probability-minded 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 multiple-choice 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 multiple-choice 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 multiple-choice 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 multiple-choice 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.)

 Start . . Text Back