Probability

1. How many ways are there to choose 5 items (with repetition) from
among 7 varieties? Explain, as precisely as possible, why we should not
factor out by the number of arrangements of the 5 choices. ( Don't prove
that the answer is wrong-we already did that- explain why it's wrong,
i.e., why we shouldn't factor out by r!) It might be helpful to think of
a smaller case: 2 choices from among 3 varieties. Following this
method, we could count 3 2/2! ways. But if you write down all 9 permutations
…

2. how many positive integers less than 1,000 are a) divisible by 7, b)
are divisible by both 7 and 11 c) have distinct digits

3. how many bit strings of length 10 contain either five consecutive 0s
or five consecutive 1s.

4. Use the product rule to show that there are 2^(2^n) different truth
tables for propositions of n variables

5. How many permutations of the letters ABCDEFGH contain:
a) the string ED
b) the strings BCA and ABF

6. How many solutions are there to the inequality x1 + x2 + x3 <= 11
where x1, x2, and x3 are non-negative integers?

7. How many different strings can be made from the letters in
MISSISSIPPI using all the letters?

8. A shelf holds 12 books in a row.  how many ways are there to choose
five books so that no two adjacent books are chosen?

9. Show that if E and F are independent events, than Ebar and Fbar are
also independent events.

10. Find the smallest number of people in a room so that the
probability that someone has a birthday today exceeds 1/2 (assume all birthdays
are equally likely and that the year has 366 days).

11. Find each of hte following probabilities when n independent
Bernoulli trials are carried out with probability of success p:
a) the probability of no successes
b) the probability of at least one success
c) the probability of at most one success
d) the probability of at least two successes