Exam 9: Counting and Probability

If six integers are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, must there be at least two integers with the property that the sum of the smaller plus the larger is 11? Why or why not? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Either describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.

A coin is loaded so that the probability of heads is 0.55 and the probability of tails is 0.45. Suppose the coin is tossed twice and the results of the tosses are independent. (a) What is the probability of obtaining exactly two heads? (b) What is the probability of obtaining exactly one head? (c) What is the probability of obtaining no heads? (d) What is the probability of obtaining at least one head?

Suppose $A$ and $B$ are events in a sample space $S$ , and $P ( A \mid B ) = 1 / 2$ and $P ( B ) = 1 / 3$ . What is $P ( A \cap B )$ ?

How many elements are in the one-dimensional array shown below? $A [ 7 ] , A [ 8 ] , \ldots , A \left[ \left[ \frac { 145 } { 2 } \right\rfloor \right]$

On each of three consecutive days the National Weather Service announces that there is a 50-50 chance of rain. Assuming that the National Weather Service is correct, what is the probability that it rains on at most one of the three days? Justify your answer. (Hint: Represent the outcome that it rains on day 1 and doesn't rain on days 2 and 3 as RNN.)