Midterm 1 Study Guide

Midterm Exam 1 will take place on Thursday October 9 during your lab section. There will be 7 problems, and they will break down as follows:

  1. (an undisclosed check on your visual intuition)
  2. rules
  3. counting
  4. counting
  5. conditional probability
  6. conditional probability
  7. discrete random variables

Below are new practice problems for each of these areas, as well as guidance about what course materials to refer to if you want more review on a particular topic.

Give your cheat sheet some TLC!

You are allowed two resources on the exam:

  • a “dumb” calculator (no wi-fi). You will not need it, but if you want it as a security blanket, be my guest;
  • both sides of one 8.5” x 11” sheet of notes, prepared by you and you alone. You can create it however you wish: handwritten, on a computer, etc.

Do not skimp on the cheat sheet! Students in past semesters have neglected their cheat sheets to their detriment. Get an early start, and give it some elbow grease.

Probability spaces

Problem 1

Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into both categories. Determine the probability that a randomly selected student is neither a genius nor a chocolate lover.

Problem 2

Suppose that \(A\) and \(B\) are mutually exclusive events for which \(P(A) = 0.3\) and \(P(B) = 0.5\). What is the probability that…

  1. either \(A\) or \(B\) occurs?
  2. \(A\) occurs but \(B\) does not?
  3. both \(A\) and \(B\) occur?
  4. Are \(A\) and \(B\) independent? Why or why not?

Problem 3

Let \(S\) be a sample space, let \(A,\, B\subseteq S\) be events, and define a new set

\[ A-B = \{x\in S:x\in A\text{ and }x\notin B\}. \]

  1. Draw a well-labeled picture of \(A\), \(B\), \(S\), and the set \(A-B\);
  2. Write down an equivalent expression for \(A-B\) that only makes use of our three basic operations: union, intersection, and complement;
  3. Prove that \(P(A-B)=P(A)-P(A\cap B)\).

Problem 4

Let \(A\), \(B\), and \(C\) be three arbitrary events, and let \(D\) denote the event that “exactly one of these three events occurs.”

  1. Draw a cartoon of \(A\), \(B\), \(C\), and \(D\);
  2. Use the three basic set operations (union, intersection, and complement) to express \(D\) in terms of \(A\), \(B\), and \(C\);
  3. Show that

\[ P(D)=P(A) + P(B) + P(C) - 2P(A \cap B) - 2P(A \cap C) - 2P(B \cap C) + 3P(A \cap B \cap C). \]

Want more review?

Counting

Problem 5

Suppose you are dealt six cards from a well-shuffled, standard deck of 52.

  1. What is the probability of getting 3 of one rank and 3 of another?
  2. What is the probability of getting 4 of one rank and 2 of another?
  3. What is the probability of getting 3 aces and 3 of another rank?
  4. What is the probability of getting 4 of one rank and 2 aces?
  5. What is the probability that all 6 cards are in the same suit?
  6. What is the probability that all 6 cards are consecutive (ie a hand with 3 hearts, 4 spades, 5 spades, 6 clubs, 7 hearts, and the 8 diamonds)? Assume that the ace can only be the high card;
  7. What is the answer to the previous question if we allow the ace to be either high or low?
  8. What is the probability of receiving the ace, king, queen, jack, 10 and 9 all in hearts?

Problem 6

Congratulations! You and two of your friends are now proud owners of brand-new library cards. In order to use your new cards, all you and your friends need to do is create four-digit PINs, with numbers \(0\) - \(9\) available (in theory) at each position. Predictably, all of you have different ideas of what an acceptable PIN is (odd or even, leading zeros or not, etc.).

  1. You are fairly chill about what you want in a PIN – all you want is an even PIN, and you’re totally fine if it starts with a zero. How many possible PINs fit your preference?
  2. One of your friends is a bit more persnickety – they want a PIN divisible by \(5\), and they cannot stand PINs starting with a zero. How many possible PINs fit this friend’s preferences?
  3. Your other friend has some more…particular preferences – they have an affinity for the number \(16\) and thus want the digits \(1\) and \(6\) to appear in that exact order at some point in the PIN. Further, they also despise PINs starting with a zero. How many possible PINs fit this friend’s preferences?

Problem 7

The United States Senate contains two senators – one senior and one junior – from each of the \(50\) states.

  1. If a committee of eight senators is selected, what is the probability that it will contain at least one of the two senators from a certain specified state?
  2. If a committee of twenty senators is selected, what is the probability it will contain \(15\) junior senators and \(5\) senior senators?
  3. What is the probability that a group of \(50\) senators selected at random will contain one senator from each state?

Want more review?

Conditional probability

Problem 8

In a standard deck of \(52\) cards, 3 (Jack, Queen, and King) of the 13 ranks are collectively known as the “face cards” of the deck. Recall that a standard deck of \(52\) cards also has \(4\) suits (Hearts, Diamonds, Clubs, and Spades).

Suppose you deal out two cards from the deck. What is the probability that they are both face cards given that they are both hearts?

Problem 9

An urn contains 5 white and 10 black balls. Four tickets labeled 1, 2, 3, and 4 reside in a box. A ticket is drawn at random from the box, and that number of balls is randomly chosen from the urn, without replacement.

  1. What is the probability that all of the balls drawn are white?
  2. If we learn that all balls drawn from the urn are white, what is the conditional probability that the #3 ticket was drawn?

Problem 10

  1. Let \(S\) be the set of all permutations of \(a\), \(b\), \(c\), together with the triples \(aaa\), \(bbb\), and \(ccc\). Imagine we draw an outcome randomly from this set. Define the event \(A_k=\{\text{the kth spot is occupied by the letter a}\}\) for \(k=1,\,2,\,3\). Compute the probabilities \(P(A_k)\), \(P(A_i\cap A_j)\), \(P(A_1\cap A_2\cap A_3)\), and comment on the independence of the three of events;

  2. Imagine we roll two fair, six-sided die. Compute \(P(A)\), \(P(B)\), \(P(C)\), \(P(A\cap B)\), \(P(A\cap C)\), \(P(B\cap C)\), and \(P(A\cap B\cap C)\) for the following three events and comment on their independence:

\[ \begin{aligned} A&=\text{first die is 1, 2, 3}\\ B&=\text{first die is 3, 4, 5}\\ C&=\text{the sum of the two rolls is 9}. \end{aligned} \]

Want more review?

Discrete random variables

Problem 11

Signore Fibonacci has created a a balanced, six-sided die whose faces are labeled, naturally, 1, 2, 3, 5, 8, 13. He gives a pair to Enzo, who rolls them repeatedly, eager to get double 13s. Let \(X\) be the number of times he has to roll the pair of dice in order to see double 13s for the first time.

  1. What is the pmf of \(X\)?
  2. What is the expected value of \(X\)?
  3. Find a closed-form expression for \(P(X\geq m)\).

Problem 12

Let \(D \sim \text{Geometric}(p)\), and fix positive integers \(n\) and \(k\). What is \(P(D = n + k \,|\, D > k)\)? Does this look familiar?

Want more review?