The second test for this course takes place in class on Wednesday, March 12. The test counts for 15% of the final grade for the course. It will be similar in format to the first test. The test will be a mix of calculation problems, short answer questions such as definitions, and longer essay-type questions. Essay questions can range from very specifc questions with a definite right answer to more general conceptual questions. I will provide calculators.
The test covers everything that we have done in class since the first test. We began by finishing up our discussion of infinity with the idea that there are different sizes of infinity and a look at Cantor's proof that the set of real numbers has strictly bigger cardinality than the set of positive integers. We then moved on to Sections 1 through 4 of Chapter 8, which are about counting and probability. Finally, we jumped to Section 10.1 for a short discussion of expected value.
Here are some of the things that you should know about for the test:
the fact that there are different sizes of infinity
Cantor's diagonalization argument: finding something that's NOT in an infinite list
probability
flipping a coin: probability of heads is 1/2; probability of tails is 1/2
rolling a 6-sided die: probability of each number is 1/6
probability of an event
= (number of outcomes for that event) / (total number of possible outcomes)
the importance of having equally likely outcomes in this definition
Example: equally likely outcomes for two coin flips are HH, HT, TH, TT
counting outcomes for multiple events like flipping several coins, rolling several dice
Also: Be able to list the outcomes
(probability an event DOES occur) = 1 - (probability it does NOT occur)
Example: (probability of getting a 6 in 4 dice rolls) =
1 - (probability of not getting a 6) = 1 - (5/6)4
examples with playing cards (52 cards in 4 suits and 13 values)
the effect of partial knowledge on probability of an event,
such as knowing at least 1 of 3 flipped coins is a head,
OR knowing that the FIRST of 3 flipped coins is a head
the Birthday problem: How likely to find two people with the same birthday
relative frequency: (number of times the event occurs) / (number of experiments done)
the difference between probability and relative frequency, and how they are related
law of large numbers: If you repeat an experiment many times, the relative frequency
of an event will tend to be close to the probability of an event
probability and infinity: what can be done by an infinite number of monkeys
extremely unlikely events are still possible (even if they will never happen in reality)
unlikely events are commonplace (but they don't usually happen to you)
counting
rules of counting: count everything; don't count anything twice
number of ways to choose k things from a set of n things:
if order matters and repetition is allowed: nk
Examples: typical "Daily Number"; drawing from a deck of cards with replacement
if order matters and repetition is NOT allowed: nPk
nPk = n×(n-1)×...×(n-k+1) [k terms]
Example: 1st, 2nd, and 3rd place in a race
if order DOESN'T matters and repetition is NOT allowed: nCk
nCk = (n×(n-1)×...×(n-k+1)) / (k×(k-1)×...×2×1)
Examples: typical "Lotto"; draw k cards from a deck of cards in the usual way
counting outcomes for one event FOLLOWED BY another event (multiply)
counting outcomes for one event OR another event (add, but don't double-count)
expected value
computing expected value from the probabilities and the values of the possible outcomes
fair games
Note that you should be able to apply these ideas to various situations, including novell ones that you haven't encountered in class.
Here are some sample questions:
1. Discuss the idea that two infinite sets can have different cardinalities. Explain what this means. What is your reaction to the idea?
2. Suppose that you have an infinite number of decks of cards. You shuffle them all together and then lay out the cards in a row: 2 of Hearts, Jack of Diamonds, 3 of Diamonds, A2 of Hearts, ... Consider the set of all such sequences of cards. Use an argument similar to Cantor's diagonalization argument to show that this set has larger cardinality than the set of positive integers.
3. Suppose that you roll three regular six-sided dice. What is the number of equally likely outcomes? What is the probability that the total on the dice is 3? What is the probability that the total on the dice is 4?
4. List all possible outcomes for tossing 4 coins, such as HHTH or THHH. (How many should there be?) What is the probability that all 4 coins are the same? What is the probability that you get equal numbers of heads and tails?
5. Suppose that someone flips five coins and tells you that the first four coins came up heads. What is the probability that the fifth coin also came up heads? Now suppose the person tells you that at least four of the five coins came up heads. What is the probability that all five coins came up heads.
6. Suppose that an order code consists of either two letters (such as AB or NQ) or a letter followed by one of the digits 0 through 9 (such as A9 or Q7). How many different order codes are there?
7. Suppose that you add two Jokers to a regular Poker deck, for a total of 54 cards. If you deal two cards from the deck, what is the probability that you get the two Jokers? What is the probability that you don't get any Jokers?
8. A "pair" means two cards of the same value. Using a regular poker deck of 52 cards, how many ways are there are drawing 5 cards from the deck without getting a pair. Based on that answer, what is the probability of getting at least one pair?
9. You repeat an experiment 1000 times. You observe that a certain event occurs in 104 of those experiments. What is the relative frequency? Do you think it's likely that the true probability of the event is 1/10? Do you think it's likely that the probability is 1/5?
10. In a horse race with 8 horses, prizes are given for 1st, for 2nd, and for 3rd place. How many different lists of winners are possible?
11. A horse breeder will pick 3 of his 8 horses to be sold. How many different ways are there of selecting the horses?
12. You pay $1 to roll a six-sided die. You get $3 if the roll is 6. You get $2 if the roll is 4 or 5. You get nothing if the roll is 1, 2, or 3. Will you play?
13. You pay $1 to roll a six-sided die. You get $3 if the roll is 6. You get $2 if the roll is 5. You get $1 if the roll is 4. You get nothing if the roll is 1, 2, or 3. Show that this game is fair.
14. A stranger stops you on the street and asks you to roll a six-sided die. He will give you as many hundred-dollar bills as the number on the die ($100 for a 1, $200 for 2, etc.). What is your expected value.
15. Problems from Section 10.1: # 14, 18, 22, 29
But remember that a significant part of the test will be essay-type questions rather than calculation problems. Also, some of the sample problems given here are harder than questions that would appear on a test.