**Stat 61 - Homework #2 (9/19/07; due 9/24/07)**

1. Prove (from the definition of independence
and axioms of probability) that if A and B are independent, then so are A and B^{C}.

2. Prove (from the definition of independence
and axioms of probability) that if A and B are independent, then so are A^{C}
and B^{C}. (Use problem 1 if it
helps.)

3. Draw one card from a standard deck of 52 cards. Consider these events:

A: the card is an ACE

C: the card is a CLUB

Use the uniform probability measure on the set of cards (that is, each card has probability 1/52 of being the card selected).

a. What is P(A)?

b. What is P(C)?

c. Are A and C independent? (Base your conclusion on the definition

of independent events.)

4. Draw two cards in succession from the same standard 52-card deck. Do NOT replace the first card before drawing the second. Consider these events:

A: The FIRST card is an ACE

C: The SECOND card is a CLUB

K: The SECOND card is a KING

Use the uniform measure on the obvious sample space (the one from HW1, with

52×52 – 52 outcomes).

a. About events A and C: What is P(A)? What is P(C)? What is P(AÇC)?

Are these events independent?

b. [CORRECTED 9-20-07] Calculate P(K) the hard way---work out the steps

in the law of total probability:

P(K) = P(first card is king) P(K|first card is king)

+ P(first card is not king) P(K | first card is not king).

c. Are A and K independent?

5. Draw one card (as in problem 3) from a 54-card deck---the usual 52 cards, plus two jokers. Let A and C be as in problem 3. Are A and C independent?

6. Draw one card from a standard deck, and consider these events:

A: the card is red

B: the card is a SPADE or HEART

C: the card is a SPADE or DIAMOND

Using the uniform probability measure on the set of cards (or the set of suits, if that suits you) show that:

A and B are independent

A and C are independent

B and C are independent

A, B, and C are NOT independent

7. (Another Bayesian update problem)

Joe has three special dice ---

one with four sides, called 1, 2, 3, 4

one standard die, with sides 1, …, 6

one with eight sides, called 1, …, 8.

He picks one at random. Define A_{4} to be the event that he
picks the 4-sided die, and define A_{6}, A_{8} similarly. These are clearly a partition of S. Initially, assign probabilities of 1/3 to each.

He rolls the selected die twice, and reports the results to you: 1, then 6. Define the event B to be this result:

B: roll 1, then roll 6.

What is the likelihood
function? ( That is, what is P( B | A_{i
}) for each of the events in the
partition A_{4}, A_{6}, A_{8} ?) [CORRECTED 9-20-07]

What are the posterior probabilities
of A_{4}, A_{6}, and A_{8} ?

COUNTING PROBLEMS:

8. (Based on text # 2.6.21.) We decided that there are 900 3-digit numbers (not allowing

leading zeros). It’s the method that counts, here; so be as systematic as you can.

a. How many of them have no repeated digits? (e.g., 202 is NOT allowed)

b. Of the ones with no repeated digits, how many are LESS THAN 289?

c. Of the ones with no repeated digits, how many are LESS THAN 298?

9. I have a collection of 10 different rare---indeed, unique---first editions, and I have decided to donate 4 of them to the Swarthmore Library. But I have to decide which ones. In how many different ways can I make that choice?

10. You have a collection of 100 different rare---indeed unique---first editions, and you have decided to donate 40 of them to the Swarthmore Library. But you have to decide which ones. In how many different ways can you make that choice?

(end)