Probability Theory Prerequisite Review

Key Topics

This review worksheet covers the main topics needed for a course focusing on probability theory. This particular sheet is intended for an introductory-upper-level course intended for math and statistics majors. There are four key prerequisite topics, as follows:

1. Integral and Differential Calculus (including some multivariable calculus, particularly double and multiple integrals)
2. Evaluating Summations (e.g., working with Taylor series)
3. Combinatorics
4. Working with Sets (Unions, Intersections, etc.)

This worksheet will provide practice examples at the level typical of a probability theory course. Use this to assess your current understanding of this material and identify areas to review further in preparation for your probability class. I highly recommend starting on this before classes start or in the first couple weeks, as professors likely will not be taking time to review!

You may also wish to review my Introductory Statistics Prerequisite Worksheet, which is intended for statistics classes with less theory and less prerequisite materials (notably calculus).

Integration Review

Evaluate the following integrals:

1. $\int_0^\infty xe^{-x^2}dx$
2. $\int_0^\infty xe^{-x}dx$
3. $\int_{-2}^2 \sqrt{4-x^2}dx$
4. $\int_{-2}^2 x\sqrt{4-x^2}dx$

Hint: problems 3 and 4 can be solved without using any trigonometric functions. (There is usually not a strong emphasis on trigonometry in these courses.)

Evaluating Summations Review

The following known sums, including some Taylor Series, are the most-often used and should be committed to memory:

$$ \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+\frac{x^2}{2} + \ldots = e^x \text{ (Taylor Series for } e^x) $$

$$ \sum_{n=0}^\infty ax^n = a +ax +ax^2+ \ldots = \frac{a}{1-x} \text{ (Infinite Geometric Series, for } |x| < 1) $$

$$ \sum_{n=0}^N ax^n = a +ax +ax^2+ \ldots + ax^N= \frac{a(1-x^N)}{1-x} \text{ (Finite Geometric Series)} $$

$$ \sum_{k=0}^n {n \choose k} a^k b^{n-k} = (a+b)^n \text{ (Binomial Formula)} $$

Recall that ${n \choose k}$, read as "$n$ choose $k$", gives the binomial coefficients:

$$ {n \choose k} = \frac{n!}{k!(n-k)!} $$

Evaluate the following sums, possibly as a simplified function of $x$. You may wish to refer to the known sums provided above.

1. $$\sum_{n=0}^{\infty} \frac{4}{3^{2n}}$$
2. $$\sum_{n=3}^{\infty} \frac{4}{3^{2n}}$$
3. $$\sum_{n=0}^{\infty} \frac{x^{2n+1}}{n!}$$
4. $$\sum_{n=1}^{10} {10 \choose n } n x^{2n-2}$$

Combinatorics Review

1. A café offers a choice of 6 different flavors of sandwiches, 6 soups, and 6 salads. How many different ways can you order a combo containing two items, assuming:

         (a) a combo can contain any two items (e.g., two BLTs is fine);

         (b) a combo CAN contain two items of the same type but not identical flavors (e.g., two different soups);

         (c) a combo CANNOT contain two items of the same type.

2. A recreational soccer club with 20 members consists of 11 women and 9 men. They break up into teams of 4 for some casual games.* How many unique ways can you create:

         (a) A team of 4 members;

         (b) A team of 4 where one is a goalie, another is a forward, another is a midfielder, and another is a backfielder;

         (c) A team of 2 women and 2 men?

*Note: My soccer knowledge is almost entirely from Ted Lasso, so please don't get upset with this somewhat unrealistic question!

Set Operations Review

Let $A$ and $B$ be subsets of $S$. For each of the pictures below, write expression using $A$, $B$, and $S$, as well as the union ($\cup$), intersection ($\cap$), and complement ($A^C$) operations that will result in a set containing exactly the shaded elements.

Challenge: For problems 2, 3, and 4, find two unique ways represent the shaded regions.

Problem 1

Problem 2

Problem 3

Problem 4

Conclusion

Answers are available here. Please comment below with any questions you have, or to discuss the problems and other prerequisite material further!