Shorter homework this week since we only meet once!

Inclusion-exclusion

1. A club consists of 10 seniors, 12 juniors, and 15 sophomores. An organizing committee of five people is chosen randomly, meaning all subsets of size 5 are equally likely.

   1. Find the probability that there are exactly 3 sophomores on the committee. (This doesn’t require inclusion-exclusion.)

   2. Find the probability that the committee has at least one senior, junior, and sophomore.

2. What is the probability that 13 random cards from a 52-card deck are void in at least one suit? (That means that there exists one suit which is not contained in the hand.)

3. A fair, six-sided die is rolled \(n\) times. This question is about the probability that you see every value at least once.

   1. Obviously, as \(n\) gets bigger, you would expect to see every value. I will bet you $100 that you don’t see every value in \(n\) rolls. Gut feeling, do you feel comfortable with \(n = 10\)?

   2. Find a formula for the probability that at least one of the six possible values never appears.

   3. How do you feel about \(n = 10\) now? What’s the smallest \(n\) that you like?

Axiomatic probability

4. Show that, for any events \(A\) and \(B\), the following statements are true.

   1. \(P(A) + B(B) - 1 \leq P(A \cap B)\)
   2. \(P(A \cap B) \leq P(A \cup B)\)
   3. \(P(A \cup B) \leq P(A) + P(B)\).

   Once you do all of these things, you just proved this:

   \[P(A) + P(B) - 1 \leq P(A \cap B) \leq P(A \cup B) \leq P(A) + P(B)\]

5. If \(A\) and \(B\) are events, then \(B - A\) (or \(B \setminus A\)) is the set of elements of \(B\) that are not in \(A\). In other words, \(B - A = B \setminus A = B \cap A^C\).

   1. Prove that, if \(A \subseteq B\), then

   \[P(B - A) = P(B) - P(A).\]

   Hint. \(B = (B - A) \cup A\)

   2. Give a sample space and events \(A, B\) such that

   \[P(B - A) \neq P(B) - P(A).\]

6. Set operations are like logical operators. Intersections are AND, unions are OR, complements are NOT.

   Given events \(A\) and \(B\), their symmetric difference, written \(A \Delta B\), is the set of all outcomes which are in \(A\) or \(B\), but not both. This is like XOR.

   Show that \[P(A \Delta B) = P(A) + P(B) - 2 P(A \cap B)\]

Probability by counting practice

7. 1. Roll a six-sided die twice. Find the probability that the dice sum to 11, and the probability that the dice sum to 12. Which is more likely?

   2. Roll a six-sided die four times. Find the probability that the dice sum to 21, and the probability that the dice sum to 22. Which is more likely?

8. A palindrome is a word that reads the same forwards as backwards.

   1. Find the probability that two randomly chosen letters form a palindrome.
   2. Find the probability that three randomly chosen letters form a palindrome.

   Which probability is larger?

9. There are 50 people in our class, and we’ll meet approximately 20 times. If you flip a coin every day you come to class, what is the probability that at least two classmates will have the same sequence of 20 flips when the term is over?