exam_i

Exam I Review

Topics

• Combinatorics including
  • Dice
  • Card Hands
• • Binomial Expansion
  • Geometric Series
  • Exponential Series
  • Logrithmic Series
• • Binomial coefficients
  • Multinomial coefficients
• • Axioms of probability
  • Conditional probability
• Baye's formula
• Independence
• Inclusion-exclusion
• • Expectation
  • Variance
• • Binomial random variable [(n,p)]
  • Poisson random variable [(\lambda)]
  • Geometric random variable [(p)]
• • Stirling's Formula
  • Integrals of probability theory

Formulas

Chapter 1: Combinatorial Analysis

Binomial Coefficient

[{n \choose i} = \frac{ n!}{ (n - i)! i!}]

where [0 \le i \le n]

Binomial Theorem

[(x + y)^n = \sum\limits_{i = 0}^{n}{ {n \choose i} x^i y^{n-i}}]

Multinomial Coefficient

For nonnegative integers [n_1, \dots, n_r] summing to [n]

[{n \choose n_1, n_2, \dots, n_r} = \frac{ n!}{ n_1! n_2! \cdots n_r!}]

is the number of division of [n] items into [r] distinct non-overlapping subgroups of sizes [n_1, n_2, \dots, n_r]

Chapter 2: Axioms of Probability

Union

• [0 \le P( A) \le 1]
• [P( S) = 1]
• For mutually exclusive events [A_i, i \ge 1]

[P\left(\bigcup\limits_{i = 1}^{\infty}{ A_i} = \sum\limits_{i = 1}^{\infty}{ P( A_i)}]

Problems

Chapter 1: Combinatorial Analysis

Problem 3

[20] workers are to be assigned to [20] different jobs, one to each job. How many different assignments are possible?

• [ 20!]
  • [20] unique jobs and [20] unique workers

Problem 7

In how many ways can 3 boys and 3 girls sit in a row?

• [6!]
  • [6] unique individuals

In how many ways can [3] boys and [3] girls sit in a row if the boys and the girls are each to sit together?

• [(2) 3! 3!]
  • [2] groups of [3] unique individuals filling [3] seats per group with [2] possible arrangements of the groups

In how many ways if only the boys must sit together?

• [(4) 3! 3!]
  • [2] groups of [3] unique individuals filling [3] seats per group with [4] possible arrangements of the groups as the groups, the group of boys having four positions between the girls.

In how many ways if no two people of the same sex are allowed to sit together?

• [(2) 3! 3!]
  • [2] groups of [3] unique individuals filling [3] seats per group with [2] possible arrangements of the groups

Problem 8

How many different letter arrangements can be made from the letters

• Fluke
  • [5!]
    • All unique letters
• Propose
  • [\frac{ 7!}{ 2! 2!} ]
    • [7] letters, [5] unique, [2] sets of [2] same latter

Problem 14

How many 5-card poker hands are there?

• [{52 \choose 5}]