Math 20, Fall 1998

*Date:* Wednesday, 27 October; 11:15 - 12:15

Instructions:

- Complete any 3 questions out of the 5 questions given.

- You may complete the 3 questions in any order.

- Each question is worth the same number of points.

- Explain all your answers. Hand in all your working notes.

- You have 60 minutes to complete the exam.

- You may use a calculator.

- 1.
- Aaron and Ayako are playing a guessing game. They have two golden
balls, and two silver balls in a large bag. Ayako picks two balls from the
large bag at random, and places them in a small bag.
- (a)
- Prove that the probability that both balls in the small bag are golden is .
- (b)
- What is the probability that the small bag contains at least one golden ball?

- (c)
- What probability should Aaron now assign to the the event that both balls in the small bag are gold?

- (d)
- What probability should Aaron now assign to the the event that the remaining ball in the small bag is gold?

- 2.
- (a)
- Use Stirling's formula to prove that the probability of getting heads times in tosses of a fair coin is approximately , when is large.

- (b)
- What is the probability that there are heads in total, given that there are heads in the first tosses?
- (c)
- What is the probability that there are heads in the first tosses, given that there are heads in total?
- (d)
- What is the probability that there are fewer than heads in total?

- 3.
- Chaim owns three boxes: a red box, a blue box, and a purple box. The
red box contains 2 gold rings and a ruby ring. The blue box contains
two gold rings and a sapphire ring, and the purple box contains a ruby
ring, and a sapphire ring.
He opens one of the boxes at random, and randomly takes one of the rings from the box.

- (a)
- What is the probability that the ring Chaim took was ruby?
- (b)
- What is the probability that the ring Chaim took was sapphire?

- (c)
- What is the probability the first ring is ruby from the red box, and the second ring is sapphire?
- (d)
- What is the probability that first ring is ruby from the purple box, and the second ring is sapphire?
- (e)
- What is the probability that one of the two rings is ruby, and the other is sapphire?

- 4.
- A candy jar contains
pieces of orange candy,
pieces of yellow
candy, and
pieces of red candy. Little Maya takes a big handful of
pieces of candy from the jar.
- (a)
- What is the probability that Maya has only orange and yellow candy?
- (b)
- What is the probability that Maya has pieces of orange candy, and pieces of yellow candy?
- (c)
- What is the probability that Maya has exactly pieces of orange candy.
- (d)
- What is the probability that Maya has exactly pieces of orange candy or exactly 5 pieces of yellow candy (or both)?

- 5.
- A fair coin is tossed three times. Let
be the number of heads that
turn up on the first two tosses. Let
be the number of heads that turn up
on all three tosses.
- (a)
- Find the joint distribution of and .
- (b)
- Find the distribution of , and the distribution of .
- (c)
- Are and independent? (Justify your answer.)
- (d)
- Find the distribution of .