Information Theory

First Assignment: due Fri. Sept. 14

  1. PI Ch. 2 #11

  2. Find the number of five card hands which form

    a. a straight (5 cards of distinct ranks differing by 1),

    b. full house flush (2 cards of one suit, 3 of another),

    c. two pair.

  3. At a party there are 6 men and 4 women among whom are 3 heterosexual couples. How many possible choices are there for the 3 couples?

  4. There are two locks on your door and you have 6 keys.

    a. What is the probability that your first choice of two keys is correct?

    b. What is the probability that you also put the keys into the correct locks on the first try?

    c. Suppose you lose one key. What is the probability that you can still get in your house (ignore the open window)? Given the lost key, answer questions a. and b.

  5. A lake has n trout and 100 of them are caught and tagged. Later 100 trout are caught and it is found that 7 of them are tagged.

    a. Compute the probability of this occurrence, assuming the 100 trout are caught all together.

    b. Compute the probability of this occurrence, assuming that after each trout is caught it is returned to the lake.

    c. Evaluate your expressions for n = 500; 1000; 10,000; 100,000 and compare.

    Given that 7 out of 100 are tagged what is your best guess as to the number of trout n ?

  6. A set of 6 distinctly colored dice are thrown. The number of possible outcomes is 6^6=46,656 ---if the color distinctions are accounted for---and $\bin{6+6-1}{6}=462$ (that's code for the binomial coefficient (6+6-1) choose 6)---if no distinction is made between the dice.

    a. We can make an even coarser measurement of the throw of the dice by only accounting for the "coincidence patterns," as in cards: for example, a pair, two pairs, a triple, ... , a sextuple. Enumerate the 11 possible coincidence patterns.

    b. There are several ways for each coincidence pattern to be realized: for example, a pair might be two 1's, 3,4,5,6. or two 1's, 2,3,5,6 (order is not important). For each coincidence pattern count the number of ways in which it can occur. The sum over all coincidence patterns should be 462.

    c. Taking account now of the color, each realization of a coincidence pattern can occur in several ways. But this is dependent only on the coincidence pattern and not on the realization. For each coincidence pattern find this number. Then add up the appropriate things to get 46,656.

  7. There is a card game called set in which each card has four attributes, and each attribute can take on one of three values as shown in the table:
    Attribute Values
    number one, two, three
    shape oval, peanut, diamond
    color red, green, purple
    shading solid, striped, empty
    There is one card for each possible combination of attribute values. A "set" is a a set of three cards such that for each attribute the cards either have all the same value, or all different values.

    a. How many cards are there?

    b. How many distinct sets are there? What is the probability that an arbitrary choice of three cards is a set?