When run, a computer program chooses the word "expensive'' with probability and otherwise chooses the word "free''. It then selects a letter uniformly at random from the word it chose and outputs that letter. Given the program outputs "e'', what is the probability that it chose the word "free''?

Which of the following probability distributions would be the best model for the number of times a standard die is rolled before a 6 occurs?

How many possible ordered triples are there such that are non-negative integers and ?

A fair coin is flipped 82 times. What is the expected number of times that a flip of tails is followed immediately by two consecutive flips of heads?

How many of the following sequences can be the degrees of the vertices of a connected simple graph (on 5 vertices)?

• 2, 1, 1, 1, 1
• 2, 2, 2, 1, 1
• 3, 3, 1, 1, 1
• 4, 3, 2, 1, 0
• 4, 4, 2, 1, 1

A tree with 81 vertices has 50 vertices of degree 1 and 30 vertices of degree 3. What is the degree of its other vertex?

How many spanning trees of the graph include neither the edge nor the edge ?

How many bridges does the graph have?

Which of the following edges could be added to the graph to produce a new graph with a closed Euler trail?

How many vertices of the graph have degree 2?