# Mid-term Examination, Discrete Probability 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?
Ayako looks in the small bag, and then teases Aaron, saying: At least one of the balls is gold. Guess what the other one is!''
(c)
What probability should Aaron now assign to the the event that both balls in the small bag are gold?
Aaron can't decide what to guess, so he asks Ayako to pick one of the balls in the small bag at random and show it to him. She closes her eyes and picks a ball. Then she looks, and says Look! This ball is gold. Now your guess is even harder to make!''
(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.
A coin is tossed one million times.
(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?
He then takes a second ring from one of the boxes (without replacing the first).
(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)